DC Optimal Power Flow Formulationand Solution Using QuadProgJ∗

Junjie Sun†and Leigh Tesfatsion‡

ISU Economics Working Paper No. 06014

Revised: 1 March 2010

Abstract

Nonlinear AC Optimal Power Flow (OPF) problems are commonly approximatedby linearized DC OPF problems to obtain real power solutions for restructured whole-sale power markets. We first present a standard DC OPF problem, which has thenumerically desirable form of a strictly convex quadratic programming (SCQP) prob-lem when voltage angles are eliminated by substitution. We next augment this standardDC OPF problem in a physically meaningful way, still retaining an SCQP form, so thatsolution values for voltage angles and locational marginal prices are directly obtainedalong with real power injections and branch flows. We then show how this augmentedDC OPF problem can be solved using QuadProgJ, an open-source Java SCQP solvernewly developed by the authors that implements the well-known dual active-set SCQPalgorithm by Goldfarb and Idnani (1983). To demonstrate the accuracy of QuadProgJ,comparative results are reported for a well-known suite of numerical QP test cases withup to 1500 decision variables plus constraints. Detailed QuadProgJ results are also re-ported for 3-node and 5-node DC OPF test cases taken from power systems texts andISO-NE/PJM training manuals.

Keywords: AC optimal power flow, DC OPF approximation, Strictly convex quadraticprogramming, Dual active-set method; Lagrangian augmentation, Java implementa-tion, QuadProgJ, AMES Market Package

JEL classifications: C61, C63, C88

∗This work has been supported in part by the National Science Foundation under Grant NSF-0527460.The authors are grateful to Deddy Koesrindartoto for dedicated collaboration on earlier phases of this project.The authors also thank Donald Goldfarb, William Hogan, Daniel Kirschen, Chen-Ching Liu, Jim McCalley,Michael J. D. Powell, Jim Price, Harold Salazar, Johnny Wong, and Tong Wu for helpful conversations ontopics related to this study.

†Starting July 9, 2007, Junjie Sun will assume the position of Financial Economist with the Office of theComptroller of the Currency, U.S. Treasury, Washington D.C.

‡Corresponding Author: Leigh Tesfatsion (tesfatsi@iastate.edu) is Professor of Economics and Mathe-matics at Iowa State University, Ames, IA 50011-1070.

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1 Introduction

The standard AC Optimal Power Flow (OPF) problem involves the minimization of totalvariable generation costs subject to nonlinear balance, branch flow, and production con-straints for real and reactive power; see Wood and Wollenberg (1996, Chpt. 13). In practice,AC OPF problems are typically approximated by a more tractable “DC OPF” problem thatfocuses exclusively on real power constraints in linearized form.

We first present a standard DC OPF problem in per unit form. This standard problemcan be represented as a strictly convex quadratic programming (SCQP) problem, that is,as the minimization of a positive definite quadratic form subject to linear constraints. AnSCQP problem can be expressed in matrix form as follows:

Minimize

f(x) =1

2xTGx + aTx (1)

with respect tox = (x1,x2, . . . ,xM)T (2)

subject to

CTeqx = beq (3)

CTiqx ≥ biq (4)

where G is an M× M symmetric1 positive definite matrix.As will be clarified below, the solution of this standard DC OPF problem as an SCQP

problem directly provides solution values for real power injections. However, solution valuesfor locational marginal prices (LMPs), voltage angles, and real power branch flows have tobe recovered indirectly by additional manipulations of these solution values.

We next show how this standard DC OPF problem can be augmented in a physicallymeaningful way, still retaining an SCQP form, so that solution values for LMPs, voltageangles, and voltage angle differences are directly recovered along with solution values for realpower injections and branch flows. We then carefully explain how this augmented SCQPproblem can be solved using QuadProgJ, an SCQP solver newly developed by the authors.QuadProgJ implements the well-known dual active-set SCQP algorithm by Goldfarb andIdnani (1983) and appears to be the first open-source SCQP solver developed completelyin Java. It is designed for the fast and efficient desktop solution of small to medium-scaleSCQP problems for research and training purposes.

More precisely, we show how the augmented DC OPF problem in SCQP form can besolved using QuadProgJ optionally coupled with an outer Java shell (DCOPFJ). This outershell automatically converts input data from standard SI units to per unit (pu), puts thispu data into the matrix form required by QuadProgJ, and then converts the pu output

1Symmetry is assumed here without loss of generality. Since xT Gx = xT GT x, the matrix G in (1) canalways be replaced by the symmetric matrix G = [G + GT ]/2.

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back into SI units. To demonstrate the accuracy of QuadProgJ, we report comparativefindings for a well-known suite of numerical QP test cases with up to 1500 decision variablesplus constraints. As a test of DCOPFJ coupled with QuadProgJ, we also present detailednumerical findings for illustrative three-node and five-node DC OPF test cases taken frompower systems texts and ISO-NE/PJM training manuals.

Section 2 presents the basic configuration of a restructured wholesale power market op-erating over an AC transmission grid, making use of a computational framework developedby the authors in previous studies. Section 3 carefully derives a standard DC OPF problemin per unit form for this wholesale power market and discusses how this standard formula-tion can be usefully augmented to enable the direct generation of solution values for LMPs,voltage angles, voltage angle differences, real power injections, and branch flows. Section 4explicitly derives and presents a complete matrix SCQP representation for this augmentedDC OPF problem. Section 5 illustrates this representation for three-node and five-node DCOPF test cases.

Section 6 then explains how the augmented DC OPF problem in SCQP form can be solvedusing QuadProgJ optionally coupled with the DCOPFJ shell. Section 7 reports comparativeQP test case results, and Section 8 presents detailed numerical findings for the three-nodeand five-node DC OPF test cases. Concluding remarks are given in Section 9. Technicalnotes on the derivation of AC power flow equations from Ohm’s Law and on the SCQPrepresentation of the standard DC OPF problem are provided in appendices.

2 Configuration of the Wholesale Power Market

Formulation of DC OPF problems for restructured wholesale power markets requires detailedstructural information about the transmission grid as well as supply offer and demand bidinformation for market participants. This section briefly but carefully describes a computa-tional framework (“AMES”) previously developed by the authors for the dynamic study ofrestructured wholesale power markets. The following Section 3 then sets out a standard DCOPF problem based on this wholesale power market framework.

2.1 Overview of the AMES Framework

In April 2003 the U.S. Federal Energy Regulatory Commission proposed a Wholesale PowerMarket Platform (WPMP) for common adoption by all U.S. wholesale power markets (FERC,2003). In a series of previous studies2 we have developed a Java framework modeling a re-structured wholesale power market operating over an AC transmission grid in accordancewith core features of the WPMP as implemented by the ISO New England in its StandardMarket Design (ISO-NE, 2003) and by the Midwest ISO in its April 2005 market initiative(MISO, 2007).

This framework – referred to as AMES 3 – includes an Independent System Operator(ISO) and a collection of bulk energy traders consisting of Load-Serving Entities (LSEs)

2See Koesrindartoto and Tesfatsion (2004), Koesrindartoto et al. (2005), and Sun and Tesfatsion (2007).3AMES is an acronym for Agent-based M odeling of E lectricity Systems.

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and Generators distributed across the nodes of the transmission grid.4 In general, multipleGenerators at multiple nodes could be under the control of a single generation company(“GenCo”), and similarly for LSEs. This control aspect is critically important to recognizefor the study of strategic trading, but it plays no role in the current study.

The AMES ISO undertakes the daily operation of the transmission grid within a two-settlement system using Locational Marginal Pricing .5 More precisely, at the beginning ofeach operating day D the AMES ISO determines hourly power commitments and LocationalMarginal Prices (LMPs)6 for the day-ahead market for day D+1 based on Generator supplyoffers and LSE demand bids (forward financial contracting). Any differences that arise duringday D+1 between real-time conditions and the contracts cleared and settled in day D for theday-ahead market for D + 1 are settled by the AMES ISO in the real-time market for D + 1at real-time LMPs. Transmission grid congestion is managed by the inclusion of congestioncomponents in LMPs.

As discussed more carefully in Sections 2.3 and 2.4 below, the current study makesthe usual empirically-based assumption that the daily demand bids of the AMES LSEsexhibit negligible price sensitivity and hence reduce to daily load profiles. In addition, it isassumed for notational simplicity that the AMES Generators submit supply offers consistingof their true marginal cost functions and true production limits (i.e., they do not makestrategic offers). In this case the optimization problem faced by the ISO for each hour ofthe day-ahead market reduces to a standard AC OPF problem requiring the minimization of(true) total variable generation costs subject to balance constraints, branch flow constraints,(true) production constraints, and given loads. As is commonly done in practice, the AMESISO approximates this nonlinear AC OPF problem by means of a DC OPF problem withlinearized constraints. The AMES ISO invokes QuadProgJ through the DCOPFJ shell inorder to solve this DC OPF problem in per unit form.

The remainder of this section explains the configuration of the AMES transmission gridand market participants.

4An Independent System Operator (ISO) is an organization charged with the primary responsibility ofmaintaining the security of a power system and often with system operation responsibilities as well. The ISOis “independent” to the extent that it does not have a conflict of interest in carrying out these responsibilities,such as an ownership stake in generation or transmission facilities within the power system. A Load-ServingEntity (LSE) is an electric utility, transmitting utility, or Federal power marketing agency that has anobligation under Federal, State, or local law, or under long-term contracts, to provide electrical power toend-use (residential or commercial) consumers or to other LSEs with end-use consumers. An LSE aggregatesindividual end-use consumer demand into “load blocks” for bulk buying at the wholesale level. A Generatoris a unit that produces and sells electrical power in bulk at the wholesale level. A node is a point on thetransmission grid where power is injected or withdrawn.

5Locational Marginal Pricing is the pricing of electrical power according to the location of its withdrawalfrom, or injection into, a transmission grid.

6A Locational Marginal Price (LMP) at any particular node of a transmission grid is the least cost ofmeeting demand for one additional unit (MW) of power at that node.

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2.2 Configuration of the AMES Transmission Grid

The AMES transmission grid is an alternating current (AC) grid modeled as a balancedthree-phase network with N ≥ 1 branches and K ≥ 2 nodes. Reactances on branches areassumed to be total reactances (rather than per mile reactances), meaning that branch lengthis already taken into account. All transformer phase angle shifts are assumed to be zero, alltransformer tap ratios are assumed to be 1, all line-charging capacitances are assumed to be0, and the temperature is assumed to remain constant over time.

The AMES transmission grid is assumed to be connected in the sense that it has noisolated components; each pair of nodes k and m is connected by a linked branch pathconsisting of one or more branches. If two nodes are in direct connection with each other, itis assumed to be through at most one branch, i.e., branch groups are not explicitly considered.However, complete connectivity is not assumed, that is, node pairs are not necessarily indirect connection with each other through a single branch.

For per unit normalization in DC OPF implementations, it is conventional to specify basevalue settings for apparent power (voltampere) and voltage.7 For the AMES transmissiongrid, the base apparent power, denoted by So, is assumed to be measured in three-phasemegavoltamperes (MVAs), and the base voltage, denoted by Vo, is assumed to be measuredin line-to-line kilovolts (kVs).

It is also assumed that Kirchoff’s Current Law (KCL) governing current flows in electricalnetworks holds for the AMES transmission grid for each hour of operation. As detailed inKirschen and Strbac (2004, Section 6.2.2.1), KCL implies that real and reactive power musteach be in balance at each node. Thus, real power must also be in balance across the entiregrid, in the sense that aggregate real power withdrawal plus aggregate transmission lossesmust equal aggregate real power injection.

In wholesale power markets restructured in accordance with FERC’s proposed WPMPmarket design (FERC, 2003), the transmission grid is overlaid with a commercial networkconsisting of “pricing locations” for the purchase and sale of electric power. A pricing locationis a location at which market transactions are settled using publicly available LMPs. Forsimplicity, it is assumed that the set of pricing locations for AMES coincides with the set oftransmission grid nodes.

2.3 Configuration of the AMES LSEs

The AMES LSEs purchase bulk power in the AMES wholesale power market in order toservice customer demand (load) in a downstream retail market. The user specifies thenumber J of LSEs as well as the location of these LSEs at various nodes of the transmissiongrid. LSEs do not engage in production or sale activities in the wholesale power market.Hence, LSEs purchase power only from Generators, not from each other.

At the beginning of each operating day D, each AMES LSE j submits a daily load profileinto the day-ahead market for day D + 1. This daily load profile indicates the real powerdemand pLj(H) that must be serviced by LSE j in its downstream retail market for each of 24

7For a detailed and careful discussion of base value determinations and per unit calculations for powersystem applications, see Anderson (1995, Chpt. 1) and Gonen (1988, Chpt. 2).

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successive hours H . In the current AMES modeling, the standard assumption is made thatthese demands are not price sensitive. One possible interpretation of this price-insensitivityassumption is that the AMES LSEs are required by retail regulations to service their loadprofiles as “native”8 load obligations, and that the profit (revenues net of costs) receivedby LSEs for servicing these load obligations is regulated to be a simple dollar mark-up overcost that is independent of the cost level. Under these conditions, LSEs have no incentiveto submit price-sensitive demand bids into the day-ahead market.

2.4 Configuration of the AMES Generators

The Ames Generators are electric power generating units. The user specifies the number Iof Generators as well as the location of these Generators at various nodes of the transmissiongrid. Generators sell power only to LSEs, not to each other.

Each AMES Generator is user-configured with technology, endowment, and learning at-tributes. Only the technology attributes are relevant for the current study. With regardto the latter, it is assumed that each Generator has variable and fixed costs of production.However, Generators do not incur no-load, startup, or shutdown costs, and they do not faceramping constraints.9

More precisely, the technology attributes assumed for each Generator i take the followingform. Generator i has minimum and maximum capacities for its hourly real power productionlevel pGi (in MWs), denoted by pL

Gi and pUGi, respectively.10 That is, for each i,

pLGi ≤ pGi ≤ pU

Gi (5)

In addition, Generator i has a total cost function giving its total costs of production perhour for each hourly production level p. This total cost function takes the form

TCi(p) = ai · p + bi · p2 + FCosti (6)

where ai ($/MWh), bi ($/MW2h), and FCosti ($/h) are exogenously given constants. Notethat TCi(p) is measured in dollars per hour ($/h). Generator i’s total variable cost functionand (prorated) fixed costs for any feasible hourly production level p are then given by

TVCi(p) = TCi(p) − TCi(0) = ai · p + bi · p2 (7)

8Native load customers for an LSE are customers whose power needs the LSE is obliged to meet bystatute, franchise, regulatory requirement, or contract.

9As is standard in economics, variable costs are costs that vary with the level of production, and fixedcosts are costs such as debt and equity obligations associated with plant investments that are not dependenton the level of production and that are incurred even if production ceases. As detailed by Kirschen andStrbac (2004, Section 4.3), the concept of no-load costs in power engineering refers to quasi-fixed costs thatwould be incurred by Generators if they could be kept running at zero output but that would vanish onceshut-down occurs. Startup costs are costs specifically incurred when a Generator starts up, and shutdowncosts are costs specifically incurred when a Generator shuts down. Finally, ramping constraints refer tophysical restrictions on the rates at which Generators can increase or decrease their outputs.

10In the current AMES modeling, the lower production limit pLGi for each Generator i is interpreted as a

firm “must run” minimum power production level. That is, if pLGi is positive, then shutting down Generator

i is not an option for the AMES ISO. Consequently, for most applications of AMES, these lower productionlimits should be set to zero.

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andFCosti = TCi(0) (8)

respectively. Finally, the marginal cost function for Generator i takes the form

MCi(p) = ai + 2 · bi · p (9)

At the beginning of each operating day D, each Generator i submits a supply offer intothe day-ahead market for use in each hour H of day D + 1. This supply offer consists ofa reported marginal cost function defined over a reported feasible production interval. Ingeneral, this supply offer could be strategic in the sense that the reported marginal costfunction deviates from Generator i’s true marginal cost function MCi(p) and the reportedfeasible production interval differs from Generator i’s true feasible production interval [pL

Gi,pU

Gi]. For the purposes of this paper, however, it can be assumed without loss of generalitythat each Generator i reports its true marginal cost function and its true feasible productioninterval.11

3 DC OPF Problem Formulation

A DC OPF problem is an approximation for an underlying AC OPF problem under sev-eral simplifying restrictions regarding voltage magnitudes, voltage angles, admittances, andreactive power. To lessen the chances of numerical instability, the variables appearing inthe resulting DC OPF problem are commonly expressed in normalized per unit (pu) valuesso that the magnitudes of these variables are more nearly equal to each other.12 In Sec-tion 3.1 we briefly but carefully outline the manner in which a standard DC OPF problemexpressed in pu values is derived from an underlying AC OPF problem expressed in standardSI (International System of Units).

Using the results of Section 3.1, we then derive in Section 3.2 a standard DC OPF prob-lem in full structural pu form for the AMES wholesale power market set out in Section 2.In particular, we show that this problem can be expressed as a strictly convex quadraticprogramming (SCQP) problem once voltage angles are eliminated by substitution from theproblem constraints. An SCQP formulation is highly desirable from the standpoint of stablenumerical solution. Unfortunately, this voltage angle substitution eliminates the nodal bal-ance constraints and hence the ability to directly generate solution values for LMPs, whichby definition are the shadow prices for the nodal balance constraints.

11Thus, the Generators’ supply offers take the form of linear upward-sloping supply curves. As detailed inSun and Tesfatsion (2007), this representation for supply offers greatly facilitates the modeling of Generatorlearning. In the actual ISO-NE and MISO wholesale power markets, generators submit their supply offers inthe form of non-decreasing step functions (MW/price blocks) defined over their feasible production intervals.However, with generator permission, the ISO uses the step points to construct smoothed offer approximations.

12As will be clarified in subsequent sections, QuadProgJ can directly accept DC OPF variable inputsexpressed in pu form so that all internal calculations are carried out in pu terms. Alternatively, as explainedin Section 6, QuadProgJ can be coupled with an outer DCOPFJ shell that automatically converts wholesalepower market variables from standard SI to per unit form prior to invoking QuadProgJ.

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Consequently, in Sections 3.3 and 3.4 we develop an alternative version of this standardDC OPF problem in pu form making use of a physically meaningful Lagrangian augmenta-tion. This augmented DC OPF problem directly generates solution values for LMPs, voltageangles, and voltage angle differences as well as real power injections and branch flows whilestill retaining a numerically desirable SCQP form.

3.1 From AC OPF to DC OPF Per Unit

Conversion of an AC OPF problem to a DC OPF approximation in per unit form requirescareful attention to variable conversions in both the problem constraints and the problemobjective function. Here we first consider constraint conversions and then take up the neededconversions for the objective function.

The key constraints in an AC OPF problem that are simplified in a DC OPF approxi-mation are the representations for real and reactive power branch flows. Let km denote abranch that connects nodes k and m with k �= m. Let Pkm (in MWs) denote the real powerbranch flow for km, and let Qkm (in MVARs) denote the reactive power branch flow for km.Let Vk and Vm denote the voltage magnitudes (in kVs) at nodes k and m, and let δk and δm

denote the voltage angles (in radians) at nodes k and m. Finally, let gkm and bkm denotethe conductance and the susceptance (in mhos) for branch km.13

Given these notational conventions, Pkm and Qkm (k �= m) can be expressed as follows:14

Pkm = V 2k gkm − VkVm[gkm cos(δk − δm) + bkm sin(δk − δm)] (10)

Qkm = − V 2k bkm − VkVm[gkm sin(δk − δm) − bkm cos(δk − δm)] (11)

The three basic assumptions used to derive a DC OPF approximation from an underlyingAC OPF problem are as follows (c.f. Kirschen and Strabac, 2004, p. 186, and McCalley,2006):

[A1] The resistance rkm for each branch km is negligible compared to the reactance xkm

and can therefore be set to 0.

[A2] The voltage magnitude at each node is equal to the base voltage Vo.

[A3] The voltage angle difference δk − δm across any branch km is sufficiently small inmagnitude so that cos(δk − δm) ≈ 1 and sin(δk − δm) ≈ [δk − δm].

Given assumption [A1], it follows that gkm = 0 and bkm = [−1/xkm], where xkm denotesthe reactance (in ohms) for branch km. Thus, Pkm = VkVm[1/xkm] sin(δk − δm) and Qkm =

13Impedance takes the complex form z = r+√−1x, where r (in ohms) denotes resistance and x (in ohms)

denotes reactance. Admittance (the inverse of impedance) then takes the complex form y = g+√−1 b, where

the conductance is given by g = r/[r2 + x2] (in mhos) and the susceptance is given by b = −x/[r2 + x2] (inmhos).

14See Appendix A for a rigorous derivation of these power flow equations from Ohm’s Law.

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V 2k [1/xkm]−VkVm[1/xkm] cos(δk−δm). Adding assumption [A2], Pkm = V 2

o [1/xkm] sin(δk−δm)and Qkm = V 2

o [1/xkm] − V 2o [1/xkm] cos(δk − δm). Finally, adding assumption [A3],

Pkm = V 2o · [1/xkm] · [δk − δm] (12)

and the reactive power branch flow Qkm in equation (11) reduces to Qkm = V 2o [1/xkm] −

V 2o [1/xkm] · 1 = 0.

As detailed in Anderson (1995, Chpt. 1) and Gonen (1988, Chpt. 2), any quantity in anelectrical network can be converted to a dimensionless pu quantity by dividing its numericalvalue by a base value of the same dimension. In power system calculations, only two basevalues are needed; and these are usually taken to be base voltage and base apparent power(voltampere). Assuming a balanced three-phase network with a base voltage Vo measuredin line-to-line kVs and a base apparent power So measured in three-phase MVAs, the baseimpedance Zo (in ohms) is specified to be

Zo = V 2o /So (13)

Given Zo, the pu reactance xkm for branch km is defined to be

xkm pu = xkm/Zo (14)

Note that xkm pu is a dimensionless quantity. Using assumption [A3], the pu susceptancebkm for branch km is given by

bkm pu = − 1/[xkm pu] (15)

Also, the pu real power branch flow Fkm for branch km is given by

Fkm = Pkm/So (16)

Now divide each side of the real power branch flow equation (12) by the base apparentpower So. Also, let Bkm denote the negative of the susceptance pu on branch km. That is,define

Bkm = − bkm pu = [1/xkm pu] (17)

It then follows from equations (13) through (17) that the real power branch flow equation(12) can be expressed in the following simple linear pu form commonly seen in power systemstextbooks:

Fkm = Bkm[δk − δm] (18)

As will be clarified below, an additional change of variables needed to express the DCOPF problem in pu terms is to everywhere divide real power quantities by base apparentpower So. Thus, for example, the real power pGi injected by each Generator i is expressedin pu terms as

PGi = pGi/So (19)

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and the real power load pLj withdrawn by each LSE j is expressed in pu terms as

PLj = pLj/So (20)

The objective function for the DC OPF problem must be expressed in pu terms as well asthe constraints. Thus, the total cost function and variable cost function defined in Section 2.4for each Generator i are expressed as a function of pu real power PGi as follows:

TCi(PGi) = Ai · PGi + Bi · P 2Gi + FCosti (21)

TVCi(PGi) = Ai · PGi + Bi · P 2Gi (22)

where Ai ($/h) and Bi ($/h) are pu-adjusted cost coefficients defined by

Ai = aiSo (23)

Bi = biS2o (24)

Note that the pu-adjusted cost functions TCi(PGi) and TVCi(PGi) are still measured indollars per hour ($/h).

Finally, as usual, one node needs to be selected as the reference node with a specifiedvoltage angle. For concreteness, we make the following assumption:

[A4] Node 1 is the reference node with voltage angle normalized to 0.

3.2 Standard DC OPF in Structural PU Form

This subsection sets out a standard DC OPF problem for the AMES wholesale power marketin full structural pu form, making use of the developments in Section 3.1. It is then seen thatthis standard problem can be expressed in numerically desirable SCQP form if the voltageangles are eliminated by substitution from the problem constraints.

For easy reference, the admissible exogenous variables and endogenous variables used inthe standard DC OPF formulation are gathered together in Tables 1 and 2, respectively.These variable definitions will be used throughout the remainder of this study.

Given the variable definitions in Tables 1 and 2, the standard DC OPF problem for theAMES wholesale power market formulated in pu terms is as follows:

MinimizeI∑

i=1

[AiPGi + BiP2Gi] (25)

with respect toPGi, i = 1, ..., I; δk, k = 1, ..., K

subject to:

Real power balance constraint for each node k = 1, ...,K:

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Table 1: DC OPF Admissible Exogenous Variables Per Unit

Variable Description Admissibility Restrictions

K Total number of transmission grid nodes K > 0

N Total number of distinct network branches N > 0

I Total number of Generators I > 0

J Total number of LSEs J > 0

Ik Set of Generators located at node k Card(∪Kk=1Ik) = I

Jk Set of LSEs located at node k Card(∪Kk=1Jk) = J

So Base apparent power (in three-phase MVAs) So ≥ 1

Vo Base voltage (in line-to-line kVs) Vo > 0

Vk Voltage magnitude (in kVs) at node k Vk = Vo, k = 1, . . . , K

PLj Real power load (pu) withdrawn by LSE j PLj ≥ 0, j = 1, . . . , J

km Branch connecting nodes k and m (if one exists) k �= m

BR Set of all distinct branches km, k < m BR �= ∅xkm Reactance (pu) for branch km xkm = xmk > 0, km ∈ BR

Bkm [1/xkm] for branch km Bkm = Bmk > 0, km ∈ BR

F Ukm Thermal limit (pu) for real power flow on km F U

km > 0, km ∈ BR

δ1 Reference node 1 voltage angle (in radians) δ1 = 0

P LGi Lower real power limit (pu) for Generator i P L

Gi ≥ 0, i = 1, . . . , I

P UGi Upper real power limit (pu) for Generator i P U

Gi > 0, i = 1, . . . , I

Ai, Bi Cost coefficients (pu adjusted) for Generator i Bi > 0, i = 1, . . . , I

FCosti Fixed costs (hourly prorated) for Generator i FCosti ≥ 0, i = 1, . . . I

MCi(P ) MCi(P ) = Ai + 2BiP = Generator i’s MC function MCi(PLGi) ≥ 0, i = 1, . . . I

Table 2: DC OPF Endogenous Variables Per Unit

Variable Description

PGi Real power injection (pu) by Generator i = 1, . . . , I

δk Voltage angle (in radians) at node k = 2, . . . , K

Fkm Real power (pu) flowing in branch km ∈ BR

PGenk Total real power injection (pu) at node k = 1, . . . , K

PLoadk Total real power withdrawal (pu) at node k = 1, . . . , K

PNetInjectk Total net real power injection (pu) at node k = 1, . . . , K

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0 = PLoadk − PGenk + PNetInjectk (26)

where

PLoadk =∑j∈Jk

PLj (27)

PGenk =∑i∈Ik

PGi (28)

PNetInjectk =∑

kmormk∈BR

Fkm (29)

Fkm = Bkm [δk − δm] (30)

Real power thermal constraint for each branch km ∈ BR:

|Fkm| ≤ F Ukm (31)

Real power production constraint for each Generator i = 1, .., I:

P LGi ≤ PGi ≤ P U

Gi (32)

Voltage angle setting at reference node 1:

δ1 = 0 (33)

As it stands, this standard DC OPF problem in pu form is a positive semi-definitequadratic programming problem. To see this, recall the general matrix form of a quadraticprogramming problem depicted in Section 1. The objective function (25) expressed in thequadratic form (1) with x = (PG1, . . . , PGI , δ1, . . . , δK)T entails a diagonal matrix G withpositive entries in its first I diagonal elements corresponding to the real power injectionsPGi but zeroes in its remaining K diagonal elements corresponding to the voltage angles δk,implying that G is a positive semi-definite matrix.

As shown in Appendix B, it is possible to use the nodal balance constraints (26) fork = 2, . . . , K together with the normalization constraint (33) to express the voltage anglevector (δ2, . . . , δK) as a linear affine function of the real power injection vector (PG1, . . . , PGI).Using this relation to everywhere eliminate the voltage angles does result in a numericallymore desirable SCQP problem. Unfortunately, this voltage angle elimination also preventsthe direct determination of solution values for LMPs since, by definition, the LMPs are theshadow prices for the nodal balance constraints.

The following subsection develops a simple physically meaningful augmentation of thestandard DC OPF objective function that permits direct generation of optimal LMPs andvoltage angle solutions while retaining a numerically desirable SCQP form.

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3.3 Augmentation of the Standard DC OPF Problem

Consider the following augmentation of the standard DC OPF objective function (25) witha soft penalty function on the sum of the squared voltage angle differences:

I∑i=1

[AiPGi + BiP2Gi] + π

[ ∑km∈BR

[δk − δm]2

](34)

As demonstrated carefully in Section 4 below, this augmentation transforms the standardDC OPF problem into an SCQP problem that can be used to directly generate solutionvalues for LMPs and voltage angles as well as real power injections and branch flows, aclear benefit. However, this augmentation also has two additional potential benefits basedon physical and mathematical considerations:

• Physical Considerations: The augmentation provides a way to conduct sensitivity ex-periments on the size of the voltage angle differences that could be informative forestimating the size and pattern of AC-DC approximation errors.

• Mathematical Considerations: The augmentation could help to improve the numericalstability and convergence properties of any applied solution method.

On the other hand, the augmentation would also seem to come with a potential cost. Specif-ically, could it cause significant distortions in the standard DC OPF solution values?

This subsection takes up each consideration in turn. The bottom line, supported byexperimental evidence, is that solution distortions appear to be practically controllable toarbitrarily small levels through appropriately small settings of the soft penalty weight π.Consequently, the benefits of augmentation would seem to strongly outweigh the costs.

3.3.1 Potential Benefits Based on Physical Considerations

The standard DC OPF problem in pu form set out in Section 3.2 requires the minimizationof total variable costs subject to a set of linearized constraints. As detailed in Section 3.1,this pu form relies on the four simplifying assumptions [A1] through [A4]. In particular, thelinear form of the branch flow constraints relies on assumption [A3] asserting that voltageangle differences across branches remain small.

Consequently, small voltage angle differences is the basis upon which a DC approximationto a true underlying AC OPF problem is formulated. Nevertheless, the standard DC OPFproblem does not constrain voltage angle differences apart from the constraints imposedthrough branch flow limits, a conceptually distinct type of constraint motivated in termsof the physical attributes of transmission lines. If the presumption of small voltage angledifferences is violated, the errors induced by reliance on a DC approximation could becomeunacceptably large.

Much remains to be done regarding how small is small enough for voltage angle differencesin order to achieve satisfactory DC OPF approximations not only for AC OPF quantitysolutions (real power injections and branch flows) but also for AC OPF price solutions (the

13

LMP at each node). We have only been able to find one study of this issue (Overbye et al.,2004) that takes both quantity and price solutions into account. The conclusions reached bythe authors on the basis of two case studies are cautiously optimistic with regard to quantitysolutions. However, as the authors note, the LMPs are determined by the binding branchflow constraints, hence small branch flow changes causing changes in the binding branch flowconstraints can have discrete and potentially large impacts on LMP solutions. For example,in the authors’ second case study, the DC approximation missed almost 50% of the bindingconstraints for the AC problem. Although many of these were “near misses,” the effects ofthese near-misses on the LMP approximations were in some cases significant.

For these reasons, it would seem prudent to pay close attention to the sizes of the voltageangle differences when undertaking DC OPF approximations to AC OPF problems. DCsolutions obtained with large voltage angle differences could diverge significantly from ACsolutions, thus giving misleading signals - particularly price signals – for the operation ofrestructured wholesale power markets.

Introducing a soft penalty function on voltage angle differences permits sensitivity checksto be conducted to determine the sensitivity of DC OPF solutions to impositions of thisprecondition for AC-DC approximation. Ideally, the DC OPF solutions obtained with suffi-ciently small soft penalty weights π should reproduce the DC OPF solutions obtained in theabsence of any soft penalty imposition, as a baseline for comparison. This is indeed seen tobe the case in the numerical π sensitivity results reported in Section 8.4.

3.3.2 Potential Benefits Based on Mathematical Considerations

As is well known, numerical stability and convergence properties of nonlinear programmingproblems with minimization (maximization) objectives can often be enhanced by increasingthe convexity (concavity) of their objective functions through suitable augmentations.

For example, the Fortran package ZQPCVX developed by Powell (1983) for convex QPminimization problems includes a simple artificial augmentation to induce strict convexity.Specifically, the matrix diagonal of the positive semi-definite quadratic form representing thenonlinear part of the objective function is augmented with positively-valued constants to in-duce positive definiteness. More generally, Shahidehpour et al. (2002, Appendix B.2) discussan entire class of artificial augmentations suitable for nonlinear programming problems withinequality constraints. The authors use versions of these augmentations on pages 288-289and elsewhere in their text to improve the convexity (hence the convergence properties) ofvarious types of optimization problems arising for electric power systems.

Although artificial augmentations can work well to ensure stability and convergence, theydo not provide meaningful sensitivity information for the physical problem at hand. Happily,as explained above, a physically meaningful augmentation is available for the standard DCOPF problem that accomplishes strict convexification of the objective function with severalimportant side benefits.

14

3.3.3 Potential Costs in Terms of Solution Distortions

In Section 8.4 we report findings for extensive tests conducted with 3-node and 5-node DCOPF problems to check the extent to which the soft penalty function augmentation affectsstandard DC OPF solution values. To briefly summarize, these findings indicate that theeffects of this augmentation on the resulting solution values are negligible for a sufficientlysmall setting of the soft penalty weight π. Moreover, no numerical instability or convergenceproblems were detected for any of the tested π values.

3.4 Augmented DC OPF in Reduced PU Form

The augmented DC OPF problem in structural pu form obtained by replacing the standardDC OPF objective function (25) by the augmented objective function (34) can be compactlyrepresented in the following reduced form:

MinimizeI∑

i=1

[AiPGi + BiP2Gi] + π

[ ∑1m∈BR

δ2m +

∑km∈BR, k≥2

[δk − δm]2

](35)

with respect toPGi, i = 1, ..., I; δk, k = 2, ..., K

subject to:

Real power balance constraint for each node k = 1, ...,K (with δ1 ≡ 0):∑i∈Ik

PGi −∑

kmormk∈BR

Bkm[δk − δm] =∑j∈Jk

PLj (36)

Real power thermal constraints for each branch km ∈ BR (with δ1 ≡ 0):

−Bkm[δk − δm] ≥ − F Ukm (37)

Bkm[δk − δm] ≥ − F Ukm (38)

Real power production constraints for each Generator i = 1, .., I:

PGi ≥ P LGi (39)

−PGi ≥ − P UGi (40)

15

4 Augmented DC OPF in SCQP Form

As a preliminary step towards a SCQP depiction for the augmented DC OPF problem inreduced pu form presented in Section 3.4, it is useful to introduce some notational conventionsto simplify the exposition. The next two subsections develop matrix representations for theobjective function and constraints. The final subsection then presents the complete SCQPdepiction in a matrix form suitable for QuadProgJ solution.

4.1 Objective Function Depiction

Consider, first, the development of a quadratic form representation for the soft penaltyfunction applied to voltage angle differences in the augmented DC OPF objective function(35). As detailed in Section 2.2, care must be taken in this representation to account for thepossible lack of direct branch connections between nodes.

To this end, define the branch connection matrix E as follows:

E =

⎡⎢⎢⎢⎢⎢⎣

0 I(1 ↔ 2) I(1 ↔ 3) · · · I(1 ↔ K)I(2 ↔ 1) 0 I(2 ↔ 3) · · · I(2 ↔ K)I(3 ↔ 1) I(3 ↔ 2) 0 · · · I(3 ↔ K)

......

.... . .

...I(K ↔ 1) I(K ↔ 2) I(K ↔ 3) · · · 0

⎤⎥⎥⎥⎥⎥⎦

K×K

(41)

where I(·) is an indicator function defined as:

I(k ↔ m) =

{1 if either km or mk ∈ BR0 otherwise

Since I(k ↔ m) = I(m ↔ k) for all k and m, it follows that Ekm = Emk for all k and m.Thus, E is a symmetric matrix.

Using this indicator function construct, the number N of distinct transmission gridbranches can be determined as follows:

N =

[K∑

k,m=1

I(k ↔ m)

]/2 (42)

If the transmission grid is completely connected, then N = K[K − 1]/2.Next, define the (voltage angle difference) weight matrix W(K) as

W(K) = 2π

⎡⎢⎢⎢⎢⎢⎣

∑k �=1 Ek1 −E12 −E13 · · · −E1K

−E21

∑k �=2 Ek2 −E23 · · · −E2K

−E31 −E32

∑k �=3 Ek3 · · · −E3K

......

.... . .

...−EK1 −EK2 −EK3 · · · ∑

k �=K EkK

⎤⎥⎥⎥⎥⎥⎦

K×K

(43)

16

For example, in the special case of a completely connected grid, the weight matrix W(K)takes the form

W(K) = 2π

⎡⎢⎢⎢⎢⎢⎣

K − 1 −1 −1 · · · −1−1 K − 1 −1 · · · −1−1 −1 K − 1 · · · −1...

......

. . ....

−1 −1 −1 · · · K − 1

⎤⎥⎥⎥⎥⎥⎦

K×K

(44)

Let δ(K)T = [δ1 . . . δK ] denote an arbitrary K-dimensional voltage angle vector with at leastone non-zero element. For K = 2 it is easily verified that

1

2δ(2)TW(2)δ(2) = π [δ1 − δ2]

2 = π

[ ∑km∈BR

[δk − δm]2

]> 0 (45)

Consequently, W(2) is a symmetric positive definite matrix. A simple induction argumenton K then establishes that W(K) is a symmetric positive definite matrix for arbitrary K ≥ 2.

Now suppose δ1 ≡ 0 and δk �= 0 for some k = 2, . . . , K, and let δT−1(K) = [δ2 . . . δK ].

Also, let Wrr(K) denote the reduced weight matrix constructed from W(K) by deleting itsfirst row and its first column as follows:

Wrr(K) = 2π

⎡⎢⎢⎢⎣

∑k �=2 Ek2 −E23 · · · −E2K

−E32

∑k �=3 Ek3 · · · −E3K

......

. . ....

−EK2 −EK3 · · · ∑k �=K EkK

⎤⎥⎥⎥⎦

(K−1)×(K−1)

(46)

It is then easily shown by a simple induction argument that

1

2δ(K)TW(K)δ(K) =

1

2δ−1(K)TWrr(K)δ−1(K) (47)

= π

[ ∑1m∈BR

δ2m +

∑km∈BR, k≥2

[δk − δm]2

]> 0

Consequently, Wrr(K) is a symmetric positive definite matrix whose quadratic form ex-

presses the soft penalty term in the augmented DC OPF objective function (35). For expo-sitional simplicity, the dimension argument K for this matrix will hereafter be suppressed.

Let the Generators’ cost attribute matrix U be defined as

U = diag[ 2B1, 2B2, · · · , 2BI ] =

⎡⎢⎢⎢⎣

2B1 0 · · · 00 2B2 · · · 0...

.... . .

...0 0 · · · 2BI

⎤⎥⎥⎥⎦

I×I

(48)

17

Recalling from Table 1 that the Generator cost coefficients Bi are assumed to be strictlypositive, it is easily seen that U is a symmetric positive definite matrix.

Finally, let the matrix G be defined by

G = blockDiag[

U Wrr

]=

[U 00 Wrr

](I+K−1)×(I+K−1)

(49)

The matrix G is clearly symmetric. Moreover, G is positive definite since its associatedquadratic form maps any vector xT = [PG1, . . . , PGI , δ2, . . . , δK ] with at least one non-zerocomponent into a strictly positive scalar. That is,

1

2xTGx =

I∑i=1

[BiP2Gi] + π

[ ∑1m∈BR

δ2m +

∑km∈BR, k≥2

[δk − δm]2

]> 0 (50)

In particular, comparing (50) with (35), it is seen that (50) provides a positive definitequadratic form representation for the nonlinear terms in the augmented DC OPF objectivefunction.

4.2 Constraint Depiction

The main factor complicating the matrix representation of the constraints for the augmentedDC OPF problem is, once again, the need to allow for the possible absence of direct branchconnections between nodes. This subsection derives special matrices to facilitate this con-straint representation.

Let the definition (17) for Bkm be extended for all k �= m as follows:

Bkm =

⎧⎨⎩

1xkm pu > 0 if km or mk ∈ BR

0 otherwise

Since xkm pu = xmk pu for all km ∈ BR, it follows that Bkm = Bmk for all k �= m. Usingthis definition for Bkm, construct the bus admittance matrix B′ as follows:

B′ =

⎡⎢⎢⎢⎢⎢⎣

∑k �=1 Bk1 −B12 −B13 · · · −B1K

−B21

∑k �=2 Bk2 −B23 · · · −B2K

−B31 −B32

∑k �=3 Bk3 · · · −B3K

......

.... . .

...−BK1 −BK2 −BK3 · · · ∑

k �=K BkK

⎤⎥⎥⎥⎥⎥⎦

K×K

(51)

The reduced bus admittance matrix B′r consisting of B′ with its first row omitted then takes

the following form:

18

B′r =

⎡⎢⎢⎢⎣

−B21

∑k �=2 Bk2 −B23 · · · −B2K

−B31 −B32

∑k �=3 Bk3 · · · −B3K

......

.... . .

...−BK1 −BK2 −BK3 · · · ∑

k �=K BkK

⎤⎥⎥⎥⎦

(K−1)×K

(52)

Let BI denote the listing of the N distinct branches km ∈ BR constituting the transmis-sion grid, lexicographically sorted as in a dictionary from lower to higher numbered nodes.Let BIn denote the nth branch listed in BI. Then the adjacency matrix A with entries of1 for the “from” node and −1 for the “to” node can be expressed as follows:

A =

⎡⎢⎢⎢⎣

J(1,BI1) J(2,BI1) · · · J(K,BI1)J(1,BI2) J(2,BI2) · · · J(K,BI2)

......

. . ....

J(1,BIN) J(2,BIN) · · · J(K,BIN )

⎤⎥⎥⎥⎦

N×K

(53)

where J(·) is an indicator function defined as:

J(i,BIn) =

⎧⎨⎩

+1 if BIn takes the form ij ∈ BR for some node j > i−1 if BIn takes the form ji ∈ BR for some node j < i

0 otherwise

for all nodes i = 1, ..., K and for all branches n = 1, ..., N

Let the reduced adjacency matrix Ar be defined as A with its first column deleted. Thus, Ar

is expressed as

Ar =

⎡⎢⎢⎢⎣

J(2,BI1) · · · J(K,BI1)J(2,BI2) · · · J(K,BI2)

.... . .

...J(2,BIN) · · · J(K,BIN)

⎤⎥⎥⎥⎦

N×(K−1)

(54)

Also, define the matrix II by

II =

⎡⎢⎢⎢⎣

I(1 ∈ I1) I(2 ∈ I1) · · · I(I ∈ I1)I(1 ∈ I2) I(2 ∈ I2) · · · I(I ∈ I2)

......

. . ....

I(1 ∈ IK) I(2 ∈ IK) · · · I(I ∈ IK)

⎤⎥⎥⎥⎦

K×I

(55)

where

I(i ∈ Ik) =

{1 if i ∈ Ik

0 if i /∈ Ik

for each i = 1, . . . , I and k = 1, . . . , K. Finally, define the matrix D to be the diagonal matrixwhose diagonal entries give the Bkm values for all distinct connected branches km ∈ BRordered as in BI. That is, with some slight abuse of notation:

19

D = diag[

D1 D2 · · · DN

]N×N

(56)

where Dn = Bkm if BIn (the nth element of BI) corresponds to branch km ∈ BR.15

4.3 The Complete SCQP Depiction

Using the notation from Sections 4.1 and 4.2, the complete SCQP depiction for the aug-mented DC OPF problem in reduced pu form set out in Section 3.4 can be expressed asfollows:

Minimize

f(x) =1

2xTGx + aTx (57)

with respect to

x =[

PG1 . . . PGI δ2 . . . δK

]T(I+K−1)×1

subject to

CTeqx = beq (58)

CTiqx ≥ biq (59)

In this SCQP depiction, the symmetric positive definite matrix G is defined as in (49), andthe vector aT is given by

aT =[

A1 · · · AI 0 · · · 0]1×(I+K−1)

The equality constraint matrix CTeq takes the form:

CTeq =

[II −B

′Tr

]K×(I+K−1)

where B′r is defined as in (52) and II is defined as in (55). The associated equality constraint

vector beq takes the form:

beq =[ ∑

j∈J1PLj

∑j∈J2

PLj · · · ∑j∈JK

PLj

]TK×1

Finally, consider the inequality constraint matrix Ciq. This matrix can be decomposedinto several column-wise submatrices corresponding to the thermal constraints (37) (callit Ct1), the thermal constraints (38) (call it Ct2), the lower production constraints (39)

15Note that the matrix H ≡ DAr maps the vector δ = (δ2, . . . , δK)T of voltage angles into the N × 1 realpower branch flow vector F ≡ Hδ. Also, as established in Appendix B, PInject = B′

rrδ, where PInjectdenotes the (K−1)×1 vector of net nodal real power injections PNetInjectk, k = 2, . . . , K, and B′

rr denotesthe matrix B′ in (51) with its first row and first column eliminated (corresponding to the reference node 1).Defining the shift matrix S ≡ H[B′

rr]−1, it follows that F = S ·PInject. Compare CAISO (2003, pp. 24-25).

20

(call it CpL), and the upper production constraints (40) (call it CpU). Note, further, thatCt1 = −Ct2 and CpL = −CpU. For easier notation, let Ct ≡ Ct1 and Cp ≡ CpL. Theinequality constraint constraint matrix Ciq can then be expressed as follows:

Ciq =[

Ct −Ct Cp −Cp

](I+K−1)×(2N+2I)

orCT

iq =[

CTt −CT

t CTp −CT

p

]T(2N+2I)×(I+K−1)

In this expression,

CTt =

[Ot −DAr

]N×(I+K−1)

where Ot is an N × I zero matrix, Ar is defined as in (54), and D is defined as in (56). Also,

CTp =

[Ip Op

]I×(I+K−1)

where Ip is an I × I identity matrix and Op is an I × (K − 1) zero matrix. Putting all theseterms together, one has:

CTiq =

⎡⎢⎢⎣

Ot −DAr

−Ot DAr

Ip Op

−Ip −Op

⎤⎥⎥⎦

(2N+2I)×(I+K−1)

Finally, the associated inequality constraint vector biq can be similarly decomposed asfollows:

biq =[

bt bt bpL bpU

]T(2N+2I)×1

where

bt =[ −F U

BI1−F U

BI2· · · −F U

BIN

]TN×1

bpL =[

P LG1 P L

G2 · · · P LGI

]TI×1

bpU =[ −P U

G1 −P UG2 · · · −P U

GI

]TI×1

21

Figure 1: A Three-Node Transmission Grid

5 Illustrative Examples

5.1 A Three-Node Illustration

Consider the special case of a completely connected transmission grid consisting of threenodes {1, 2, 3}, three Generators, and three LSEs, with Generator k and LSE k located atnode k for k = 1, 2, 3. This three-node case is depicted in Figure 1.

For this three-node case, the augmented DC OPF problem set out in Section 3.4 reducesto the following form:

Minimize3∑

i=1

[AiPGi + BiP2Gi] + πδ2

2 + πδ23 + π[δ2 − δ3]

2 (60)

with respect toPG1, PG2, PG3, δ2, δ3

subject to:

Real power balance constraint for each node k = 1, ..., 3:

PG1 + B12δ2 + B13δ3 = PL1 (61)

PG2 − [B12 + B23]δ2 + B23δ3 = PL2 (62)

PG3 + B23δ2 − [B13 + B23]δ3 = PL3 (63)

22

Real power thermal constraints for each branch km ∈ BR:

B12δ2 ≥ −F U12 (64)

B13δ3 ≥ −F U13 (65)

−B23δ2 + B23δ3 ≥ −F U23 (66)

−B12δ2 ≥ −F U12 (67)

−B13δ3 ≥ −F U13 (68)

B23δ2 − B23δ3 ≥ −F U23 (69)

Real power production constraints for each Generator i = 1, ..., 3:

PG1 ≥ P LG1 (70)

PG2 ≥ P LG2 (71)

PG3 ≥ P LG3 (72)

−PG1 ≥ −P UG1 (73)

−PG2 ≥ −P UG2 (74)

−PG3 ≥ −P UG3 (75)

Using the notation introduced in Section 4, the SCQP depiction for this three-node caseis as follows:

Minimize

f(x) =1

2xTGx + aTx (76)

with respect tox = [PG1, PG2, PG3, δ2, δ3]

T(5×1) (77)

subject toCT

eqx = beq (78)

CTiqx ≥ biq (79)

23

where

G =

⎡⎢⎢⎢⎢⎣

2B1 0 0 0 00 2B2 0 0 00 0 2B3 0 00 0 0 4π −2π0 0 0 −2π 4π

⎤⎥⎥⎥⎥⎦

(5×5)

aT =[

A1 A2 A3 0 0](1×5)

CTeq =

⎡⎣ 1 0 0 B12 B13

0 1 0 −[B12 + B23] B23

0 0 1 B23 −[B13 + B23]

⎤⎦

(3×5)

beq =[

PL1 PL2 PL3

]T(3×1)

CTiq =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 B12 00 0 0 0 B13

0 0 0 −B23 B23

0 0 0 −B12 00 0 0 0 −B13

0 0 0 B23 −B23

1 0 0 0 00 1 0 0 00 0 1 0 0−1 0 0 0 00 −1 0 0 00 0 −1 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12×5)

biq =[ −F U

12 −F U13 −F U

23 −F U12 −F U

13 −F U23 P L

G1 P LG2 P L

G3 −P UG1 −P U

G2 −P UG3

]T(12×1)

Note that the first six rows in matrix CTiq correspond to thermal inequality constraints and

the next six rows correspond to power production inequality constraints.

5.2 A Five-Node Illustration

Now consider a five-node case for which the transmission grid is not completely connected.As depicted in Figure 2, let five Generators and three LSEs be distributed across the gridas follows: Generators 1 and 2 are located at node 1; LSE 1 is located at node 2; Generator3 and LSE 2 are located at node 3; Generator 4 and LSE 3 are located at node 4; andGenerator 5 is located node 5.

This information implies the following structural configuration for the transmission grid:

24

Figure 2: A Five-Node Transmission Grid

K = 5; I = 5; J = 3;

I1 = {G1, G2}, I2 = {∅}, I3 = {G3}, I4 = {G4}, I5 = {G5};J1 = {∅}, J2 = {LSE1}, J3 = {LSE2}, J4 = {LSE3}, J5 = {∅};∑

j∈J1

PLj = 0,∑j∈J2

PLj = PL1 ,∑j∈J3

PLj = PL2,∑j∈J4

PLj = PL3,∑j∈J5

PLj = 0;

The distinct directly-connected node pairs are (1,2), (1,4), (1,5), (2,3), (3,4), (4,5), whichimplies that the number of distinct transmission grid branches is N = 6. The branchconnection matrix E can be written as follows:

E =

⎡⎢⎢⎢⎢⎣

0 1 0 1 11 0 1 0 00 1 0 1 01 0 1 0 11 0 0 1 0

⎤⎥⎥⎥⎥⎦

5×5

(80)

The weight matrix W and its reduced form Wrr are

W = 2π

⎡⎢⎢⎢⎢⎣

3 −1 0 −1 −1−1 2 −1 0 0

0 −1 2 −1 0−1 0 −1 3 −1−1 0 0 −1 2

⎤⎥⎥⎥⎥⎦

5×5

(81)

25

Wrr = 2π

⎡⎢⎢⎣

2 −1 0 01 2 −1 00 −1 3 −10 0 −1 2

⎤⎥⎥⎦

4×4

(82)

The Generators’ cost attribute matrix U is:

U = diag[

2B1 2B2 2B3 2B4 2B5

]5×5

(83)

The matrix B′ and its reduced form B′r are as follows:

B′ =

⎡⎢⎢⎢⎢⎣

B12 + B14 + B15 −B12 0 −B14 −B15

−B21 B21 + B23 −B23 0 00 −B32 B32 + B34 −B34 0

−B41 0 −B43 B41 + B43 + B45 −B45

−B51 0 0 −B54 B51 + B54

⎤⎥⎥⎥⎥⎦

5×5

(84)

B′r =

⎡⎢⎢⎣

−B21 B21 + B23 −B23 0 00 −B32 B32 + B34 −B34 0

−B41 0 −B43 B41 + B43 + y45 −B45

−B51 0 0 −B54 B51 + B54

⎤⎥⎥⎦

4×5

(85)

With a slight abuse of notation, the ordered list BI of distinct transmission grid branchescan be denoted as follows:

BI = [(1, 2), (1, 4), (1, 5), (2, 3), (3, 4), (4, 5)]T6×1 (86)

The adjacency matrix A with entries of 1 for the “from” node and −1 for the “to” node canbe expressed as

A =

⎡⎢⎢⎢⎢⎢⎢⎣

1 −1 0 0 01 0 0 −1 01 0 0 0 −10 1 −1 0 00 0 1 −1 00 0 0 1 −1

⎤⎥⎥⎥⎥⎥⎥⎦

6×5

(87)

and its reduced form Ar can be expressed as

Ar =

⎡⎢⎢⎢⎢⎢⎢⎣

−1 0 0 00 0 −1 00 0 0 −11 −1 0 00 1 −1 00 0 1 −1

⎤⎥⎥⎥⎥⎥⎥⎦

6×4

(88)

26

The matrix II takes the form

II =

⎡⎢⎢⎢⎢⎣

1 1 0 0 00 0 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

⎤⎥⎥⎥⎥⎦

5×5

(89)

Finally, the matrix D takes the form

D = diag[

B12 B14 B15 B23 B34 B45

]6×6

(90)

Using the above developments, the SCQP depiction for the augmented DC-OPF problemfor this five-node case can be expressed as follows:

Minimize

f(x) =1

2xTGx + aTx

with respect to

x =[

PG1 PG2 PG3 PG4 PG5 δ2 δ3 δ4 δ5

]T9×1

subject to

CTeqx = beq

CTiqx ≥ biq

where the input matrices and vectors G, aT, CTeq, beq, CT

iq, and biq take the followingexplicit forms:

G = blockDiag[

U Wrr

]9×9

aT =[

A1 A2 A3 A4 A5 0 0 0 0]1×9

CTeq =

[II −B

′Tr

]5×9

whereB′

r is defined as in (85)

II is defined as in (89)

beq =[

0 PL1 PL2 PL3 0]T5×1

27

CTiq =

[CT

t −CTt CT

p −CTp

]T22×9

where

CTt =

[Ot −DAr

]6×9

Ot = 6 × 5 zero matrix

Ar is defined as in (88)

D is defined as in (90)

CTp =

[Ip Op

]5×9

Ip = 5 × 5 identity matrix

Op = 5 × 4 zero matrix

biq =[

bt bt bpL bpU

]T22×1

where

bt =[ −F U

12 −F U14 −F U

15 −F U23 −F U

34 −F U45

]T6×1

bpL =[

P LG1 P L

G2 P LG3 P L

G4 P LG5

]T5×1

bpU =[ −P U

G1 −P UG2 −P U

G3 −P UG4 −P U

G5

]T5×1

6 QuadProgJ Input/Output and Logical Progression

The matrix form of a general SCQP problem is presented in Section 1. QuadProgJ acceptsinput in this matrix form. In particular, QuadProgJ can be directly used to solve any DCOPF problem expressed in this matrix form whether the DC OPF variables are expressed instandard SI units (e.g. ohms, megawatts,...) or in normalized per unit (pu) terms.

On the other hand, to help ensure numerical stability, it is customary when solving DCOPF problems to carry out all internal calculations in pu terms so that variables have roughlythe same order of magnitude. The pu solution output is then often converted back into SIunits for easier readability.

Consequently, to facilitate the application of QuadProgJ to DC OPF problems, we havedeveloped an optional outer Java shell for QuadProgJ, referred to as DCOPFJ, that carriesout the following data manipulations: (a) accepts DC OPF input data in SI units and

28

converts it to pu; (b) uses this pu input data to form the SCQP matrix and vector expressionsrequired by QuadProgJ; (c) invokes QuadProgJ to solve this SCQP problem; (d) convertsthe resulting pu solution output back into SI units.

Consider the augmented DC OPF problem set out in Section 3.4. The required inputdata for this problem, expressed in SI units, can be schematically depicted as follows:

(SI gridData, SI genData, SI lseData)

where

SI gridData = (SI nodeData, SI branchData)

SI nodeData = (K, π)

SI branchData = (BI, pUBI1

...pUBIN

, X ohms)

SI genData = (I, I1...IK , a1...aI , b1...bI , pLG1...p

LGI , pU

G1...pUGI)

SI lseData = (J, J1...JK ,∑j∈J1

pLj ...∑j∈JK

pLj)

This SI input data is fed into DCOPFJ along with a base apparent power value So and abase voltage value Vo. The DCOPFJ shell first uses the base values to transform the SI inputdata into pu terms. Using the pu notation introduced in Section 3.1, this pu input data canbe schematically depicted as follows:

(pu gridData, pu genData, pu lseData)

where

pu gridData = (pu nodeData, pu branchData)

pu nodeData = (K, π)

pu branchData = (BI, F UBI1

...F UBIN

, X pu)

pu genData = (I, I1...IK , A1...AI , B1...BI , P LG1...P

LGI , P U

G1...PUGI)

pu lseData = (J, J1...JK ,∑j∈J1

PLj ...∑j∈JK

PLj)

DCOPFJ next uses this pu input data to form the matrices and vectors (G, a,Ceq,beq,Ciq,biq)as detailed in Section 4.3. It then feeds these matrix and vector components into the Quad-ProgJ solver to obtain a solution in pu terms. This pu solution can be expressed in thefollowing vector form:

(P ∗G1...P

∗GI , δ∗2 ...δ

∗K , λ∗

eq, λ∗iq) (91)

In this output vector, (P ∗G1...P

∗GI) denotes the vector of optimal pu real power production

commitments in the day-ahead market for Generators i = 1, . . . , I, and (δ∗2 ...δ∗K) denotes the

vector of optimal voltage angles (in radians) at nodes k = 2, . . . , K (omitting the referencenode 1 where δ1 is normalized to 0). The solution vector for the Lagrange multipliers

29

corresponding to the equality constraints is contained in the K × 1 vector λ∗eq. Since each of

these multipliers is a shadow price corresponding to a nodal balance constraint in pu form,λ∗

eq provides the vector of Locational Marginal Prices (LMPs) in pu form.The solution vector for the Lagrange multipliers corresponding to the inequality con-

straints is contained in the (2N + 2I) × 1 vector λ∗iq. These multipliers provide valuable

additional sensitivity information, including “flow gate” prices (in pu) measuring the opti-mal cost reductions that would result from relaxations in the branch flow constraints.

Finally, the pu solution (91) is fed back into DCOPFJ for conversion into SI units forreporting purposes. Recalling from Section 3.1 that pu real power terms are obtained fromSI real power terms (in MWs) by dividing through by the base apparent power So, this SIoutput data can be schematically depicted as follows:

(p∗G1...p∗GI , δ∗2 ...δ

∗K , λ∗

eq/So, λ∗iq/So) , (92)

where the voltage angles δ∗k are still reported in radians.In summary, the overall logical flow of the QuadProgJ program can be depicted as follows:

So, Vo, SI gridData, SI genData, SI lseData

⇓DCOPFJ

⇓Conversion of SI Input to Per Unit Matrix Form

⇓G, a,Ceq,beq,Ciq,biq

⇓QuadProgJ

⇓P ∗

G1...P∗GI , δ∗2 ...δ

∗K , λ∗

eq, λ∗iq

⇓DCOPFJ

⇓Conversion of Per Unit Output to SI Units

⇓p∗G1...p

∗GI , δ∗2 ...δ

∗K , λ∗

eq/So, λ∗iq/So

30

7 QP Test Results for QuadProgJ

7.1 Overview

QuadProgJ is a stand-alone open-source Java SCQP solver newly developed by the authors.QuadProgJ implements the well-known dual active-set SCQP method developed by Goldfarband Idnani (1983) in a numerically stable way by utilizing Cholesky decomposition and QRfactorization. For ease of use, QuadProgJ modifies the original Goldfarb and Idnani methodto permit the direct explicit imposition of equality as well as inequality constraints.

As with any dual active-set SCQP method (Fletcher, 1987, pp. 243-245), QuadProgJproceeds as follows. In the first iteration all problem constraints are ignored and the tentativeoptimal solution is taken to be the unconstrained minimum (which exists by strict convexityof the objective function). A test is then made to see if any of the original problem constraintsare violated. If so, one of these violated constraints is selected and added to the “active set,”i.e., the set of constraints to be imposed as equalities. A new optimal solution is thengenerated, subject to the active set of constraints, and again a test is made to see if anyof the original problem constraints are violated. If so, one is selected to be added to theactive set (and a test is made to see if any of the previously active constraints should nowbe relaxed). A new constrained optimal solution is then generated. This process continuesuntil no violated original problem constraints are found.

Compared to other QP methods, such as interior point and primal active-set QP methods,a dual active-set SCQP method such as QuadProgJ has two major advantages. First, it hasa well-defined starting point: namely, the unconstrained minimum of the objective function.In contrast, other types of methods typically have to guess or search for a “good” startingpoint, which can be very costly in terms of actual computing time. Second, since there areonly finitely many distinct permutations of the inequality constraints to determine which ifany are active (binding), and each activated constraint leads to an increase in the currentobjective function value, a dual active-set SCQP method is guaranteed to terminate in afinite number of steps. Infinite looping can arise with other types of methods for reasonssuch as a flat starting point.

On the downside, however, QuadProgJ has two main limitations. First, QuadProgJrequires the QP objective function to be a strictly convex function.16 Second, QuadProgJdoes not incorporate sparse matrix techniques. Consequently, it is not designed to handlelarge-scale problems for which speed and efficiency of computations become critical limitingfactors.

In this section a well-known repository of QP test cases is used to demonstrate theaccuracy of QuadProgJ for small to medium-scale QP problems.

16See Section 3.3.2 for brief notes on Lagrangian augmentation methods that can be used to induce strictconvexity for convex QP objective functions. Solution algorithms designed to handle non-strictly convex QPproblems have been developed by Boland (1997), Fletcher (1987), Powell (1983), and Stoer (1992).

31

7.2 QP Test Case Results

The accuracy of QuadProgJ has been tested on a collection of small to medium-sized SCQPminimization problems included in the QP test case repository prepared by Maros andMeszaros (1997).17 For each of these problems, the solution value for the minimized ob-jective function obtained by QuadProgJ is compared against the corresponding solutionvalue reported for BPMPD, a well-known proprietary C-language QP solver implementingan interior-point algorithm.18

The general structure of these SCQP test cases is given in Table 3, along with the reportedBPMPD solution values. Corresponding test case results for QuadProgJ are then reportedin Table 4.19 Specifically, Table 4 reports the relative difference (RD) between the minimumobjective function value f ∗ = f(x∗) obtained by QuadProgJ and the minimum objectivefunction value fBPMPD attained by BPMPD, where

RD ≡ f ∗ − fBPMPD

|fBPMPD| (93)

To help ensure a fair comparison, f ∗ has been rounded off to the same number of decimalplaces as fBPMPD.

In addition, Table 4 reports tests conducted to check whether all equality and inequal-ity constraints are satisfied at the minimizing solution x∗ obtained by QuadProgJ. Moreprecisely, for any given SCQP test case, the equality constraints take the form

CTeqx = beq (94)

and the inequality constraints take the form

CTiqx ≥ biq (95)

Let TNEC denote the total number of equality constraints for this test case (i.e. the rowdimension of CT

eq), and let TNIC denote the total number of inequality constraints for thistest case (i.e. the row dimension of CT

iq). Also, let x∗ denote the solution obtained byQuadProgJ for this test case.

The equality constraints for each SCQP test case are checked by computing the EqualityConstraint Error (ECE) for this test case, defined to be the TNEC × 1 residual vector

ECE ≡ Ceqx∗ − beq (96)

17Detailed input and output data for the SCQP test cases are available online at:http://www.sztaki.hu/˜meszaros/public ftp/qpdata/. Most of the test cases are in standard QPSformat. The QPS format is an extension of the MPS format, which is the industrial standard format forlinear programming test cases.

18See the BPMPD web site for detailed information. URL: http://www.sztaki.hu/˜meszaros/bpmpd/19All of the results reported in Table 4 for QuadProgJ were obtained from runs on a laptop PC: namely,

a Compaq Presario 2100 running under Windows XP SP2 (mobile AMD Athlon XP 2800+ 2.12 GHz, 496MB of RAM). The reported results for the BPMPD solver are taken from Maros and Meszaros (1997), whodo not identify the hardware platform on which the BPMPD solver runs were made.

32

Table 3: SCQP Test Cases: Structural Attributes and BPMPD Solution Values

NAMEa TNDb TNECc TNICd TNCe TNf fBPMPDg

DUAL1 85 1 170 171 256 3.50129662E-02DUAL2 96 1 192 193 289 3.37336761E-02DUAL3 111 1 222 223 234 1.35755839E-01DUAL4 75 1 150 151 226 7.46090842E-01DUALC1 9 1 232 233 242 6.15525083E+03DUALC5 8 1 293 294 302 4.27232327E+02HS118 15 0 59 59 74 6.64820452E+02HS21 2 0 5 5 7 -9.99599999E+01HS268 5 0 5 5 10 5.73107049E-07HS35 3 0 4 4 7 1.11111111E-01HS35MOD 3 0 5 5 8 2.50000001E-01HS76 4 0 7 7 11 -4.68181818E+00KSIP 20 0 1001 1001 1021 5.757979412E-01QPCBLEND 83 43 114 157 240 -7.84254092E-03QPCBOEI1 384 9 971 980 1364 1.15039140E+07QPCBOEI2 143 4 378 382 525 8.17196225E+06QPCSTAIR 467 209 696 905 1372 6.20438748E+06S268 5 0 5 5 10 5.73107049E-07MOSARQP2 900 0 600 600 1500 -0.159748211E+04

aCase name (in QPS format), see Maros and Meszaros (1997) for a detailed description of the QPS formatbTotal number of decision variablescTotal number of equality constraintsdTotal number of inequality constraintseTotal number of constraints (equality and inequality). TNC=TNEC+TNICfTotal number of decision variables and constraints (problem size). TN=TND+TNCgMinimizing solution value obtained by the BPMPD solver on an unknown hardware platform

Table 4 reports the mean and maximum of the absolute values of the components of thisECE vector for each SCQP test case, denoted by Mean|ECE| and Max|ECE| respectively.

Similarly, the inequality constraints for each SCQP test case are checked by computingthe Inequality Constraint Error (ICE), defined to be the TNIC × 1 residual vector

ICE ≡ Ciqx∗ − biq (97)

Table 4 reports the Number of Violated Inequality Constraints (NVIC) for each SCQP testcase, meaning the number of negative components in this ICE vector.

Based on the results presented in Table 4, it appears that the QuadProgJ solver hasan accuracy level slightly better than the BPMPD solver for small to medium-sized SCQPproblems, that is, for SCQP problems for which the total number (TN) of decision variablesplus constraints is less than 1500. This conclusion is supported by the observation that,for each of these test cases, the minimized objective function value f ∗ = f(x∗) obtained byQuadProgJ either equals or is strictly smaller than the corresponding minimized objective

33

Table 4: QuadProgJ Test Case Results

NAME Mean|ECE|a Max|ECE|b NVICc f*d RDe

DUAL1 0.0 0.0 0 3.50129657E-2 -1.42804239E-8DUAL2 0.0 0.0 0 3.37336761E-2 0.0DUAL3 6.66E-16 6.66E-16 0 1.35755837E-1 -1.47323313E-8DUAL4 2.11E-15 2.11E-15 0 7.46090842E-1 0.0DUALC1 2.40E-12 2.40E-12 0 6.15525083E+3 0.0DUALC5 5.33E-15 5.33E-15 0 4.27232327E+2 0.0HS118 NAf NA 0 6.64820450E+2 -3.00833103E-9HS21 NA NA 0 -99.96 -1.00040010E-9HS268 NA NA 0 -5.47370291E-8 -1.09550926HS35 NA NA 0 1.11111111E-1 0.0HS35MOD NA NA 0 2.50000000E-1 -4.00000009E-9HS76 NA NA 0 -4.68181818 0.0KSIP NA NA 0 5.75797941E-1 0.0QPCBLEND 5.66E-16 8.94E-15 0 -7.84254307E-3 -2.74145844E-7QPCBOEI1 2.05E-6 9.58E-6 0 1.15039140E+7 0.0QPCBOEI2 3.42E-6 1.37E-5 0 8.17196224E+6 -1.22369628E-9QPCSTAIR 4.34E-7 6.01E-6 0 6.20438745E+6 -4.83528799E-9S268 NA NA 0 -5.47370291E-8 -1.09550926MOSARQP2 NA NA — OOMEg —

aMean of the absolute values of the components of ECE (Equality Constraint Error)bMaximum of the absolute values of the components of ECEcTotal number of violated inequality constraintsdMinimum objective function value as computed by QuadProgJeRelative difference [f*-fBPMPD]/|fBPMPD| between the QuadProgJ and BPMPD solution values forthe minimized objective function. A negative value indicates QuadProgJ improves on BPMPD.

fNA indicates “Not Applicable,” meaning there are no constraints of the indicated type.gOut-of-Memory Error indicated by a run-time Java Exception: java.lang.OutOfMemoryError

function value fBPMPD obtained by BPMPD, with no indication that the QuadProgJsolution x∗ violates any equality or inequality constraints.20

Even in cases in which QuadProgJ improves on the BPMPD solution, however, therelative difference between the two solutions tends to be extremely small, generally on theorder of 10−7. The only exceptions are the two cases HS268 and S268 where QuadProgJappears to improve significantly on the BPMPD solver. HS268 and S268 are relatively simpleSCQP minimization problems subject only to inequality constraints, none of which turns outto be binding at the optimal solution. Why the interior-point BPMPD solver appears todegrade in accuracy on such problems is unclear.

All in all, QuadProgJ either matches or improves on the BPMPD solutions for all of thesmall and medium-sized SCQP test cases reported in Table 4, i.e. for all of the test cases

20Maros and Meszaros (1997) do not provide constraint checks for the BPMPD solutions reported in theirrepository.

34

for which TN (the total number of constraints plus decision variables) is less than 1500.Since the BPMPD solver has been in use since 1998, and is considered to have a proven highquality for solving QP problems, this finding suggests that QuadProgJ is at least as accuratea solver as BPMPD for SCQP problems of this size.

As noted previously, however, QuadProgJ is not designed for large-scale problems. Thetest results presented in Table 4 show that an out-of-memory error was triggered when anattempt was made to use QuadProgJ to solve test case MOSARQP2 with size TN = 1500.Whether this finding reflects an intrinsic limitation of QuadProgJ or is simply a desktoplimitation that could be ameliorated by installing additional memory or by using a differenthardware platform is an issue requiring further study.

8 DC OPF Test Case Results

8.1 Overview

In this section, QuadProgJ is used to solve illustrative three-node and five-node DC OPFtest cases taken from power systems texts and ISO-NE/PJM training manuals.

Each of these DC OPF test cases is solved by invoking QuadProgJ through the outer Javashell DCOPFJ. Specifically, given SI input data and base apparent power and base voltagevalues as detailed in Section 6, DCOPFJ invokes QuadProgJ to solve for optimal real powerinjections, real power branch flows, voltage angles, LMPs, total variable costs, and variousother output values. In particular, DCOPFJ automates the conversion of SI data to pu formfor internal calculations and forms all needed matrix/vector representations.

These illustrative DC OPF test cases raise intriguing economic issues concerning the ISOoperation of wholesale power markets in the presence of constraints on branch flows andproduction levels. The information content of LMPs in relation to these constraints is ofparticular interest. For the study at hand, however, these test cases are simply used to illus-trate concretely the capability of QuadProgJ to generate detailed DC OPF solution values.The systematic study and interpretation of DC OPF solutions generated via QuadProgJ inthe context of carefully constructed experimental designs is left for future studies.

The section concludes with a separate reporting of sensitivity results for the soft penaltyweight π > 0 for both the three-node and five-node DC OPF test cases. These resultsdemonstrate that the DC OPF solution values depend on the value of π in the expectedway. The magnitude of the summed voltage angle differences is inversely related to themagnitude of π. However, for sufficiently small π the sensitivity of the DC OPF solutionvalues to further decreases in π becomes negligible. Moreover, no numerical instability orconvergence problems were detected at any of these tested π values.

8.2 Three-Node Test Results

Table 5 provides SI input as well as base apparent power and base voltage levels So andVo for a day-ahead wholesale power market operating over a three-node transmission gridas depicted in Figure 1. The daily (24 hour) load distribution for the day-ahead market is

35

Figure 3: 24 Hour Load Distribution for a 3-Node Case

depicted in Figure 3. Note that LSE 2 and LSE 3 have identical load profiles. In addition,Generator 1 has the least expensive cost (as measured by the cost attributes a and b), andGenerator 2’s cost is between the cost of Generator 1 and Generator 3. This input data isadopted from Tables 8.2-8.4 (p. 297) in Shahidehpour et al. (2002).21

Tables 6-7 present DC OPF solution results in SI units for this day-ahead market for 24successive hours. Specifically, Table 6 reports solution values for the real power injection p∗Gi

for each Generator i, the optimal voltage angle δ∗k for each non-reference node k, and theLMP (λ∗

eqk/So) for each node k. Table 7 reports solution values for the twelve inequalityconstraint multipliers, the first six corresponding to thermal limits on branch flows and thefinal six corresponding to lower and upper bounds on production levels. Also reported inthis table are the solution values for real power branch flows.

As seen in Table 7, the branch flow multipliers are all zero. This means there are nobinding branch flow constraints, hence no branch congestion that would force higher-costGenerators to be dispatched prior to lower-cost Generators. Consequently, one would expectto see Generator 1 used to meet load demand as much as possible. Generator 2 should onlyproduce more than its minimum production level when the load demand is so high that itexceeds the maximum production level of Generator 1, and Generator 3 should only produce

21Unfortunately, Shahidehpour et al. (2002) do not provide corresponding DC OPF solution values thatcould be used to compare against QuadProgJ solution values. Their focus is on the derivation of unitcommitment schedules subject to additional security constraints that help to ensure reliability in the eventof line outages.

36

more than its minimum production level when load demand is so high that it exceeds themaximum production level of Generator 2.

The solution results reported in Table 6 are consistent with these theoretical predictions.Examining the output columns for p∗G1, p∗G2, and p∗G3, one sees the following pattern. Forthe low-demand off-peak hours (i.e. hours 02-08), Generator 1 is supplying as much of theload as possible; Generator 2 and Generator 3 are producing at their minimum productionlevels (10 MWs and 5 MWs, respectively). In contrast, for the high-demand peak hours (i.e.hours 01 and 09-24), Generator 1 is producing at its maximum production level (200 MWs)and Generator 2’s production exceeds its minimum production level (10Mws). This clearlyshows that dispatch priority is being based on cost attributes.

The column “minTVC” in Table 6 reports minimized total variable cost for each hoursummed across all Generators. For the three-node example at hand, which has three Gen-erators,

minTVC =3∑

i=1

[ai · p∗Gi + bi · p∗2Gi] (98)

As expected, minTVC changes hour by hour to reflect changes in the corresponding load;compare the daily load profile depicted in Figure 3.

Another important consistency check follows from the observation, made above, that allof the branch flow multipliers in Table 7 are zero, indicating the absence of any branchcongestion. The absence of branch congestion implies that the LMPs should be the sameacross all nodes for each hour. This is verified by output columns LMP1, LMP2, and LMP3

in Table 6.Finally, Table 7 reports six multiplier values corresponding to six real power production

constraints, two (lower and upper) for each of the three Generators. These multiplier valuesare entirely consistent with the results in Table 6. For example, the multiplier value associ-ated with the minimum (lower) production level for Generator 3 is strictly positive for eachhour, which is consistent with the result in Table 6 that Generator 3 is scheduled to produceat its minimum production level (5 MWs) for each hour.

8.3 Five-Node Test Results

Table 8 presents SI input data for a day-ahead wholesale power market operating over afive-node transmission grid as depicted in Figure 2.22 The daily (24 hour) load distributionin SI units for the day-ahead market is depicted in Figure 4. Tables 9-13 report the optimalsolution values in SI units for real power production levels, voltage angles, LMP values, min-imum total variable cost, inequality constraint multipliers, and branch flows for 24 successivehours in the day-ahead market.

In contrast to the three-node case, this five-node case exhibits branch congestion. Specif-ically, branch congestion occurs between node 1 and node 2 (and only these nodes) in each

22The transmission grid, reactances, and locations of Generators and LSEs for this 5-node example areadopted from an example developed by John Lally (2002) for the ISO-NE that is now included in trainingmanuals prepared by the ISO-NE (2007) and PJM (2007). The general shape of the LSE load profiles isadopted from a 3-node example presented in Shahidehpour et al. (2002, pp. 296-297).

37

Figure 4: 24 Hour Load Distribution for a 5-Node Case

of the 24 hours. This can be verified directly by column P12 in Table 13, which shows thatthe real power flow P12 on branch km = 12 is at its upper thermal limit (250 MWs) foreach hour. It can also be verified indirectly by column “12” in Table 11, which shows thatthe thermal inequality constraint multiplier for branch km = 12 is positively valued for eachhour, indicating a binding constraint. The direct consequence of this branch congestion isthe occurrence of widespread LMP separation, i.e. the LMP values differ across all nodesfor each hour. This can be verified by examining output columns LMP1-LMP5 in Table 10.

Examining this LMP data more closely, it is seen that LMP2 and LMP3 (the LMPs fornodes 2 and 3) exhibit a sharp change in hour 18, increasing between hour 17 and hour 18 byabout 100% and then dropping back to “normal” levels in hour 19 and beyond. Interesting,this type of sudden spiking in LMP values is also observed empirically in MISO’s DynamicLMP Contour Map23 for real-time market prices, which is updated every five minutes.

This rather dramatic LMP peaking in hour 18 can be traced to several factors. First, asseen in Figure 4, the load profile for each LSE peaks at hour 18. Second, when solving theDC OPF problem to meet the high load in hour 18, the ISO has to take into considerationthe maximum production limit for Generator 3 as well as the thermal inequality constraintbetween node 1 and node 2. Both of these constraints turn out to be binding. Specifically,as seen in Table 9, Generator 3 is dispatched in hour 18 at its maximum production limit(520 MWs); and, as seen in Table 13, the real power flow in branch km = 12 is at its upper

23http://www.midwestmarket.org/page/LMP%20Contour%20Map%20&%20Data

38

limit (250 MWs) for all 24 hours. Given the configuration of the transmission grid, to meetthe hour 18 peak load the ISO is forced to back down (relative to hour 17) the less expensiveproduction of Generators 1 and 2 and to use instead the more expensive production of the“peaker” Generator 4.

After the peak hour 18, the load returns to lower levels. The ISO is then able to dispatchGenerator 1 and Generator 2 at their more “normal” levels, with Generator 1 at its upperproduction limit, and to avoid dispatching any production from generation 4; note fromTable 8 that the minimum production level of Generator 4 is 0. Furthermore, the LMPsdrop back to their more normal levels after hour 18.

8.4 Π Sensitivity Test Results

Sensitivity tests were conducted to check the extent to which the solution values reportedin Sections 8.2 and 8.3 for the three-node and five-node DC OPF test cases depend on thespecific choice of the soft penalty weight π.

For the three-node case, a separate solution set was generated for each of the followingfive π values: 100, 10, 1, 0.1 and 0.01. These five solution sets are reported in Tables 14-18.These solution results show that decreasing the value of π over the tested range from 100 to0.01 had little impact on the resulting solution values. The only perceptible changes at thereported precision level (four decimal places) were in the LMP values in their second andhigher decimal places. Moreover, the LMP values stabilized through two decimal places (i.e.to values rounded off to pennies) once π decreased to the level 1.0.

Tables 19 and 20 report the sum of squared voltage angle differences for the three-nodeand five-node DC OPF test cases as the soft penalty weight π is decreased in value from 100to 0.01. As can be seen, the maximum absolute differences (MaxAD) between these sumsare extremely small: namely, about 10−15 in magnitude for the three-node case and about10−7 in magnitude for the five-node case. In the three-node case, any change in these sumsin response to the changes in the value of π are below visibility in the reported data. In thefive-node case, however, the sums are seen to increase slightly as the value of π decreases,which is the expected result of decreasing the penalty attached to the sum.

Also as expected, the sum of squared voltage angle differences increases with an increasein nodes from three to five for each tested value of π. This suggests that a researchermight need to tailor the value of π to the problem at hand in order to achieve a desireddegree of smallness for voltage angle differences. In addition, in some situations it might bedesirable to introduce individual weights on the voltage angle differences instead of using acommon weight π, e.g. in order to represent transmission grid losses. This could easily beaccomplished by a simple respecification of the weight matrix W in (43).

9 Concluding Remarks

Restructured electricity markets are extraordinarily complex. For example, restructuredwholesale power markets in the U.S. typically involve spot and forward energy marketsoperated by ISO/RTOs over AC transmission grids subject to congestion effects. As reported

39

by Joskow (2006, Table 1), over 50% of the generation capacity in the U.S. is now operatingunder this market design, and other regions of the U.S. are moving towards this form oforganization.

The complexity of restructured electricity markets essentially forces electricity researchersto resort to computational methods of analysis. Unfortunately, much of the software cur-rently available for computational electricity modeling is commercial and hence proprietary.This restricts the ability of electricity researchers to publish self-sufficient studies permittingfull access to implementation.

A key stumbling block to developing open-source software for general academic researchinto restructured electricity markets is the need to model the AC/DC optimal power flow(OPF) problems that must repeatedly be solved by ISO/RTO operators in order to generatedaily unit commitment and dispatch schedules, as well as locational marginal prices (LMPs),for both spot and forward energy markets. Developing algorithms for the successful solutionof optimization problems involving mixed collections of equality and inequality constraints,even when specialized to quadratic objective functions (as in DC OPF approximations toAC OPF problems), is a daunting task full of pitfalls for the unwary.

This study reports the development of QuadProgJ, an open-source plug-and-play Javasolver for strictly convex quadratic programming (SCQP) problems that can be applied tostandard DC OPF problems for research and training purposes. QuadProgJ implements thewell-known dual active-set SCQP algorithm developed by Goldfarb and Idnani (1983). Theaccuracy of QuadProgJ is demonstrated by means of comparative results for a well-knownsuite of QP test problems with up to 1500 decision variables plus constraints.

In addition, this study proposes a physically meaningful augmentation of the standardDC OPF problem that permits the direct generation of solution values for LMPs, voltageangles, and voltage angle differences together with real power injections and branch flows.Three-node and five-node test cases are used to demonstrate how QuadProgJ, coupled with aJava outer shell DCOPFJ, can be used to directly generate complete solution values for thisaugmented DC OPF problem. In particular, DCOPFJ automates the SI/pu conversion andmatrix/vector representation of all needed input data for this augmented DC OPF problem.

Appendix A: Derivation of Power Flow Branch Equations

Recall from Section 3.1 that equations for the flow of real and reactive power in any trans-mission grid branch km (k �= m) are depicted as follows:

Pkm = V 2k gkm − VkVm[gkm cos(δk − δm) + bkm sin(δk − δm)] (99)

Qkm = − V 2k bkm − VkVm[gkm sin(δk − δm) − bkm cos(δk − δm)] (100)

This appendix provides a rigorous derivation of these equations from Ohm’s Law.

A.1 Preliminary: The Relationship Between Impedance and Admittance

Using standard notational conventions, the impedance z on a transmission grid branch isexpressed as

40

z = r + jx (impedance = resistance +√−1 reactance) (101)

and the admittance y on a transmission grid branch is expressed as

y = g + jb (admittance = conductance +√−1 susceptance) (102)

Since y = 1/z, it follows that

y =1

z=

1

r + jx=

r

r2 + x2+ j

−x

r2 + x2(103)

Thus,

g =r

r2 + x2

b =−x

r2 + x2

A.2 Derivation of Equations (99) and (100)

The following derivation24 is based on Gonen (1988, (2.4)). Boldface letters denote complexvariables while letters in normal font denote real variables. Also, the following trigonometricidentities will be used in this derivation:

cos(α − β) = cos α cos β + sin α sin β

sin(α − β) = sin α cos β − cos α sin β

Let km denote any transmission grid branch, and let Skm (in MVA) denote the complexpower flowing in this branch. This complex power can be represented as

Skm = Pkm + jQkm = VkI∗km (104)

wherej =

√−1

Vk = Vk cos δk + jVk sin δk

Ikm = Current (in Amperes) on branch km

I∗km = Complex conjugate of Ikm

By Ohm’s Law in AC settings,

24Recall from Section 2.2 that all transformer tap ratios are assumed to be 1, and all transformer phaseangle shifts and line-charging capacitances are assumed to be 0. For an alternative derivation of the powerflow equations that permits general settings for these variables, see Hogan (2002, Appendix).

41

Ikm =Vk −Vm

zkm(105)

where the impedance zkm on branch km can be expressed as

zkm = rkm + jxkm

The complex conjugate of the impedance zkm∗ is then written as

z∗km = rkm − jxkm

Consequently, Skm can be written as:

Skm = VkV∗

k −V∗m

z∗km

= [Vk cos δk + jVk sin δk][Vk cos δk − jVk sin δk] − [Vm cos δm − jVm sin δm]

rkm − jxkm

=[V 2

k cos2 δk + V 2k sin2 δk] − Vk[cos δk + j sin δk]Vm[cos δm − j sin δm]

rkm − jxkm

=V 2

k − VkVm [(cos δk cos δm + sin δk sin δm) + j(sin δk cos δm − cos δk sin δm)]

rkm − jxkm

=V 2

k − VkVm[cos(δk − δm) + j sin(δk − δm)]

rkm − jxkm(Let θ = δk − δm)

=[rkm + jxkm]V 2

k − [rkm + jxkm]VkVm[cos θ + j sin θ]

[rkm + jxkm][rkm − jxkm]

=rkmV 2

k − VkVm[rkm cos θ − xkm sin θ]

r2km + x2

km

+ jxkmV 2

k − VkVm[rkm sin θ + xkm cos θ]

r2km + x2

km

=(V 2

k gkm − VkVm[gkm cos θ + bkm sin θ])

+ j(−V 2

k bkm − VkVm[gkm sin θ − bkm cos θ])

= Pkm + jQkm

Hence, we can infer that (99) and (100) hold.

Appendix B: Expressing DC OPF Voltage Angles as a Linear AffineFunction of Real Power Injections

This section establishes that the vector δ of non-reference voltage angles (δ2, . . . , δK)T in thestandard DC OPF problem in pu form presented in Section 3.2 can be expressed as a linearaffine function of the vector of real power injections.

42

The basic equations to consider are the real power nodal balance constraints (26) fork = 2, . . . , K together with the normalization δ1 = 0 imposed on the reference node voltageangle δ1 by constraint (33). When the nodal balance constraint for any node k ≥ 2 isexpressed solely in terms of voltage angles, real power injections, and real power loads, ittakes the following form:∑

i∈Ik

PGi −∑

kmormk∈BR

Bkm[δk − δm] =∑j∈Jk

PLj (106)

This collection of nodal balance constraints for k = 2, . . . , K can equivalently be expressedin matrix form as follows:

PNetInject = B′rrδ (107)

where PNetInject denotes the (K − 1) × 1 vector of net nodal real power injectionsPNetInjectk for nodes k = 2, . . . , K, and B′

rr denotes the bus admittance matrix B′ in(51) with its first row and first column eliminated (corresponding to the reference node 1).For concrete illustration, equation (107) for the 5-node test case presented in Section 5.2takes the following specific form:

⎡⎢⎢⎣

0 − PL1

PG3 − PL2

PG4 − PL3

PG5 − 0

⎤⎥⎥⎦ =

⎡⎢⎢⎣

B21 + B23 −B23 0 0−B32 B32 + B34 −B34 0

0 −B43 B41 + B43 + B45 −B45

0 0 −B54 B51 + B54

⎤⎥⎥⎦

4×4

⎡⎢⎢⎣

δ2

δ3

δ4

δ5

⎤⎥⎥⎦(108)

Since the matrix B′rr is invertible by construction, we have the following relationship

between the voltage angles and the net nodal power injections:

δ = [B′rr]

−1PNetInject (109)

In terms of the 5-node test case, equation (109) takes the following form:

⎡⎢⎢⎣

δ2

δ3

δ4

δ5

⎤⎥⎥⎦ =

⎡⎢⎢⎣

B21 + B23 −B23 0 0−B32 B32 + B34 −B34 0

0 −B43 B41 + B43 + B45 −B45

0 0 −B54 B51 + B54

⎤⎥⎥⎦

−1

4×4

⎡⎢⎢⎣

0 − PL1

PG3 − PL2

PG4 − PL3

PG5 − 0

⎤⎥⎥⎦(110)

The net nodal power injection vector PNetInject can be further decomposed into alinear affine function of the real power injection vector PG = (PG1, . . . , PGI)

T as follows:

PNetInject = RPG + β (111)

where R is a (K − 1) × I matrix and β is a (K − 1) × 1 vector defined as follows

43

R =

⎡⎢⎢⎢⎣

I(1 ∈ I2) I(2 ∈ I2) · · · I(I ∈ I2)I(1 ∈ I3) I(2 ∈ I3) · · · I(I ∈ I3)

......

. . ....

I(1 ∈ IK) I(2 ∈ IK) · · · I(I ∈ IK)

⎤⎥⎥⎥⎦

(K−1)×I

(112)

where

I(i ∈ Ik) =

{1 if i ∈ Ik

0 if i /∈ Ik

β =[ −∑

j∈J2PLj −∑

j∈J3PLj · · · −∑

j∈JKPLj

]T

(K−1)×1(113)

Again using the 5-node test case for concrete illustration, we can write out equation (111)as

⎡⎢⎢⎣

0 − PL1

PG3 − PL2

PG4 − PL3

PG5 − 0

⎤⎥⎥⎦ =

⎡⎢⎢⎣

0 0 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

⎤⎥⎥⎦

⎡⎢⎢⎢⎢⎣

PG1

PG2

PG3

PG4

PG5

⎤⎥⎥⎥⎥⎦ +

⎡⎢⎢⎣

−PL1

−PL2

−PL3

0

⎤⎥⎥⎦ (114)

Finally, combining (109) and (111), we see that it is possible to solve explicitly for thevoltage angle vector δ as a linear affine function of the real power injection vector PG:namely,

δ = R∗PG + ν (115)

where R∗ = [B′rr]

−1R and ν = [B′rr]

−1β.

44

References

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Boland, Natashia L. (1997), “A Dual-Active-Set Algorithm for Positive Semi-Definite QuadraticProgramming,” Mathematical Programming 78, 1-27.

CAISO (2003), “Locational Marginal Pricing (LMP) Study: Analysis of Cost-Based Differ-entials,” Market Design 2002 , CAISO Market Operations, California ISO, February 4release.

FERC (2003), Notice of White Paper , U.S. Federal Energy Regulatory Commission, IssuedApril 28.

Fletcher, Roger (1987), Practical Methods of Optimization, Second Edition, John Wiley &Sons, New York.

Goldfarb, Donald, and Ashok Udhawdas Idnani (1983), “A Numerically Stable Dual Methodfor Solving Strictly Convex Quadratic Programs,” Mathematical Programming 27, 1-33.

Gonen, Turan (1988), Modern Power System Analysis, Wiley-Interscience, John Wiley &Sons, Inc., New York.

Hogan, William W. (2002), “Financial Transmission Rights Formulations,” Report, Cen-ter for Business and Government, John F. Kennedy School of Government, HarvardUniversity, Cambridge, MA.

ISO-NE (2007), Home Page, ISO New England, Inc., accessible at http://www.iso-ne.com/

ISO-NE (2003), Standard Market Design Reference Guide, ISO New England, Inc., 45pp.Available at http://www.iso-ne.com/smd/

Joskow, Paul (2006), “Markets for Power in the United States: An Interim Assessment,”The Energy Journal , Vol. 27(1), 1-36.

Kirschen, Daniel S., and Goran Strbac (2004), Fundamentals of Power System Economics ,John Wiley & Sons, Ltd.

Koesrindartoto, Deddy, and Leigh Tesfatsion (2004), “Testing the Economic Reliabilityof FERC’s Wholesale Power Market Platform: An Agent-Based Computational Ap-proach,” Energy, Environment, and Economics in a New Era, Proceedings of the 24thUSAEE/IAEE North American Conference, Washington, D.C., July 8-10.

Koesrindartoto, Deddy, Junjie Sun, and Leigh Tesfatsion (2005), “An Agent-Based Compu-tational Laboratory for Testing the Economic Reliability of Wholesale Power MarketDesigns,” Proceedings , Vol. 1, IEEE Power Engineering Society General Meeting, SanFrancisco, CA, June 2005, pp. 931-936.

45

Lally, John (2002), “Financial Transmission Rights: Auction Example,” Section 6, M-06Financial Transmission Rights Draft 01-10-02, ISO New England, Inc., January.

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Overbye, Thomas J., Xu Cheng, and Yan Sun (2004), “A Comparison of the AC and DCPower Flow Models for LMP Calculations,” Proceedings , 37th Hawaii InternationalConference on System Sciences. http://csdl.computer.org/comp/proceedings/hicss/

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Sun, Junjie and Leigh Tesfatsion (2007), “Dynamic Testing of Wholesale Power MarketDesigns: An Agent-Based Computational Approach”, Economics Working Paper No.06025, Iowa State University, revised June.Available at: http://www.econ.iastate.edu/tesfatsi/DynTestAMES.JSLT.pdf

Wood, Allen J., and Bruce F. Wollenberg (1996), Power Generation, Operation, and Con-trol , Second Edition, John Wiley & Sons, Inc., New York.

46

Table 5: DC OPF Input Data in SI Units for Three-Node Case

Base Valuesa

So Vo

100 10

Kb πc

3 0.05

BranchFrom To lineCapd xe

1 2 55 0.201 3 55 0.402 3 55 0.25

GenID atNode FCost a b pMinf pMaxg

1 1 14 10.6940 0.00463 20 2002 2 21 18.1000 0.00612 10 1503 3 11 37.8896 0.01433 5 20

LSEID atNode L-00h L-01 L-02 L-03 L-04 L-05 L-06 L-07

1 1 132.66 122.4 115.62 112.2 108.84 110.52 112.2 119.042 2 44.22 40.8 38.54 37.4 36.28 36.84 37.4 39.683 3 44.22 40.8 38.54 37.4 36.28 36.84 37.4 39.68ID atNode L-09 L-10 L-11 L-12 L-13 L-14 L-15 L-16

1 1 136.02 149.64 153.06 154.74 153.06 149.64 147.96 147.962 2 45.34 49.88 51.02 51.58 51.02 49.88 49.32 49.323 3 45.34 49.88 51.02 51.58 51.02 49.88 49.32 49.32ID atNode L-17 L-18 L-19 L-20 L-21 L-22 L-23 L-24

1 1 154.74 170.04 163.26 161.52 159.84 156.42 147.96 137.762 2 51.58 56.68 54.42 53.84 53.28 52.14 49.32 45.923 3 51.58 56.68 54.42 53.84 53.28 52.14 49.32 45.92

aThe base voltage Vo is measured in kVs and the base apparent power So is measured in MVAs. Forillustrative purposes, So and Vo are chosen such that the base impedance Zo satisfies Zo = V 2

o /So = 1bTotal number K of nodescSoft penalty weight π for voltage angle differencesdUpper limit PU

km (in MWs) on magnitude of real power flow in branch kmeReactance xkm (in ohms) for branch kmfLower limit pL

Gi (in MWs) on real power production for Generator igUpper limit pU

Gi (in MWs) on real power production for Generator ihL-H: Load (in MWs) for hour H, where H=00,02,...,23

47

Table 6: DC OPF Solution Results in SI Units for Three-Node Case

Hour p∗G1 p∗G2 p∗G3 δ∗2a δ∗3 LMP1

b LMP2 LMP3 minTVCc

01 200.0 16.1 5.0 -0.0799 -0.1095 18.30 18.30 18.30 2993.9502 189.0 10.0 5.0 -0.0808 -0.1048 12.44 12.44 12.44 2724.3303 177.7 10.0 5.0 -0.0752 -0.0979 12.34 12.34 12.34 2565.1204 172.0 10.0 5.0 -0.0724 -0.0944 12.29 12.29 12.29 2485.7005 166.4 10.0 5.0 -0.0696 -0.0910 12.23 12.23 12.23 2408.2706 169.2 10.0 5.0 -0.0710 -0.0927 12.26 12.26 12.26 2446.9107 172.0 10.0 5.0 -0.0724 -0.0944 12.29 12.29 12.29 2485.7008 183.4 10.0 5.0 -0.0780 -0.1014 12.39 12.39 12.39 2645.1309 200.0 21.7 5.0 -0.0741 -0.1077 18.37 18.37 18.37 3097.9010 200.0 44.4 5.0 -0.0506 -0.1002 18.64 18.64 18.64 3527.1311 200.0 50.1 5.0 -0.0447 -0.0983 18.71 18.71 18.71 3636.9012 200.0 52.9 5.0 -0.0418 -0.0974 18.75 18.75 18.75 3691.1113 200.0 50.1 5.0 -0.0447 -0.0983 18.71 18.71 18.71 3636.9014 200.0 44.4 5.0 -0.0506 -0.1002 18.64 18.64 18.64 3527.1315 200.0 41.6 5.0 -0.0535 -0.1011 18.61 18.61 18.61 3473.5116 200.0 41.6 5.0 -0.0535 -0.1011 18.61 18.61 18.61 3473.5117 200.0 52.9 5.0 -0.0418 -0.0974 18.75 18.75 18.75 3691.1118 200.0 78.4 5.0 -0.0154 -0.0890 19.06 19.06 19.06 4193.6419 200.0 67.1 5.0 -0.0271 -0.0927 18.92 18.92 18.92 3968.9820 200.0 64.2 5.0 -0.0301 -0.0937 18.89 18.89 18.89 3911.8321 200.0 61.4 5.0 -0.0330 -0.0946 18.85 18.85 18.85 3856.8522 200.0 55.7 5.0 -0.0389 -0.0965 18.78 18.78 18.78 3745.5123 200.0 41.6 5.0 -0.0535 -0.1011 18.61 18.61 18.61 3473.5124 200.0 24.6 5.0 -0.0711 -0.1067 18.40 18.40 18.40 3152.03

aVoltage angle solutions δ∗k are reported in radiansbLocational marginal price, LMPk = λ∗

eqk/So for each node kcMinimized total variable cost

48

Table 7: DC OPF Solution Results in SI Units for Three-Node Case - Inequality ConstraintMultipliers and Real Power Branch Flows

Branch km multipliers Production constraint multipliers Branch FlowHour 12 13 23 21 31 32 P L

G1 P LG2 P L

G3 P UG1 P U

G2 P UG3 P12 P13 P23

01 0 0 0 0 0 0 0 0 19.74 5.75 0 0 39.96 27.38 11.8402 0 0 0 0 0 0 0 5.78 25.59 0 0 0 40.40 26.20 9.6003 0 0 0 0 0 0 0 5.88 25.69 0 0 0 37.61 24.47 9.0704 0 0 0 0 0 0 0 5.94 25.75 0 0 0 36.20 23.60 8.8005 0 0 0 0 0 0 0 5.99 25.80 0 0 0 34.82 22.74 8.5406 0 0 0 0 0 0 0 5.96 25.77 0 0 0 35.51 23.17 8.6707 0 0 0 0 0 0 0 5.94 25.75 0 0 0 36.20 23.60 8.8008 0 0 0 0 0 0 0 5.83 25.64 0 0 0 39.02 25.34 9.3409 0 0 0 0 0 0 0 0 19.67 5.82 0 0 37.06 26.92 13.4210 0 0 0 0 0 0 0 0 19.39 6.10 0 0 25.31 25.05 19.8311 0 0 0 0 0 0 0 0 19.32 6.17 0 0 22.36 24.58 21.4412 0 0 0 0 0 0 0 0 19.29 6.20 0 0 20.91 24.35 22.2313 0 0 0 0 0 0 0 0 19.32 6.17 0 0 22.36 24.58 21.4414 0 0 0 0 0 0 0 0 19.39 6.10 0 0 25.31 25.05 19.8315 0 0 0 0 0 0 0 0 19.42 6.06 0 0 26.76 25.28 19.0416 0 0 0 0 0 0 0 0 19.42 6.06 0 0 26.76 25.28 19.0417 0 0 0 0 0 0 0 0 19.29 6.20 0 0 20.91 24.35 22.2318 0 0 0 0 0 0 0 0 18.97 6.51 0 0 7.71 22.25 29.4319 0 0 0 0 0 0 0 0 19.11 6.38 0 0 13.56 23.18 26.2420 0 0 0 0 0 0 0 0 19.15 6.34 0 0 15.06 23.42 25.4221 0 0 0 0 0 0 0 0 19.18 6.31 0 0 16.51 23.65 24.6322 0 0 0 0 0 0 0 0 19.25 6.24 0 0 19.46 24.12 23.0223 0 0 0 0 0 0 0 0 19.42 6.06 0 0 26.76 25.28 19.0424 0 0 0 0 0 0 0 0 19.63 5.86 0 0 35.56 26.68 14.24

P U12 P U

13 P U23

55 55 55

49

Table 8: DC OPF Input Data in SI Units for Five-Node Case

Base ValuesSo Vo

100 10

K π5 0.05

BranchFrom To lineCap x

1 2 250 0.02811 4 150 0.03041 5 400 0.00642 3 350 0.01083 4 240 0.02974 5 240 0.0297

GenID atNode FCost a b pMin pMax

1 1 16 14 0.005 0 1102 1 19 15 0.006 0 1003 3 28 25 0.010 0 5204 4 10 30 0.012 0 2005 5 24 10 0.007 0 600

LSEID atNode L-01 L-02 L-03 L-04 L-05 L-06 L-07 L-08

1 2 350.00 322.93 305.04 296.02 287.16 291.59 296.02 314.072 3 300.00 276.80 261.47 253.73 246.13 249.93 253.73 269.203 4 250.00 230.66 217.89 211.44 205.11 208.28 211.44 224.33ID atNode L-09 L-10 L-11 L-12 L-13 L-14 L-15 L-16

1 2 358.86 394.80 403.82 408.25 403.82 394.80 390.37 390.372 3 307.60 338.40 346.13 349.93 346.13 338.40 334.60 334.603 4 256.33 282.00 288.44 291.61 288.44 282.00 278.83 278.83ID atNode L-17 L-18 L-19 L-20 L-21 L-22 L-23 L-24

1 2 408.25 448.62 430.73 426.14 421.71 412.69 390.37 363.462 3 349.93 384.53 369.20 365.26 361.47 353.73 334.60 311.533 4 291.61 320.44 307.67 304.39 301.22 294.78 278.83 259.61

50

Table 9: DC OPF Solution Results in SI Units for Five-Node Case - Optimal Real PowerProduction Levels and Optimal Voltage Angles (in Radians)

Hour p∗G1 p∗G2 p∗G3 p∗G4 p∗G5 δ∗2 δ∗3 δ∗4 δ∗501 110.00 13.87 332.53 0.00 443.59 -0.0702 -0.0595 -0.0394 0.016402 110.00 13.44 269.41 0.00 437.54 -0.0702 -0.0624 -0.0385 0.016203 110.00 13.16 227.70 0.00 433.54 -0.0702 -0.0643 -0.0379 0.016104 110.00 13.01 206.66 0.00 431.52 -0.0703 -0.0653 -0.0376 0.016005 110.00 12.87 185.99 0.00 429.53 -0.0703 -0.0662 -0.0373 0.016006 110.00 12.95 196.33 0.00 430.53 -0.0702 -0.0658 -0.0375 0.016007 110.00 13.01 206.66 0.00 431.52 -0.0703 -0.0653 -0.0376 0.016008 110.00 13.30 248.75 0.00 435.55 -0.0703 -0.0633 -0.0382 0.016209 110.00 14.01 353.20 0.00 445.58 -0.0703 -0.0585 -0.0397 0.016410 110.00 14.58 437.00 0.00 453.61 -0.0702 -0.0546 -0.0409 0.016611 110.00 14.73 458.03 0.00 455.63 -0.0702 -0.0536 -0.0412 0.016712 110.00 14.80 468.37 0.00 456.62 -0.0702 -0.0532 -0.0413 0.016713 110.00 14.73 458.03 0.00 455.63 -0.0702 -0.0536 -0.0412 0.016714 110.00 14.58 437.00 0.00 453.61 -0.0702 -0.0546 -0.0409 0.016615 110.00 14.51 426.67 0.00 452.62 -0.0702 -0.0551 -0.0407 0.016616 110.00 14.51 426.67 0.00 452.62 -0.0702 -0.0551 -0.0407 0.016617 110.00 14.80 468.37 0.00 456.62 -0.0702 -0.0532 -0.0413 0.016718 2.07 0.00 520.00 108.88 522.63 -0.0702 -0.0488 -0.0300 0.022219 107.35 6.12 520.00 0.00 474.13 -0.0702 -0.0507 -0.0418 0.017520 110.00 15.08 510.08 0.00 460.63 -0.0702 -0.0512 -0.0419 0.016821 110.00 15.01 499.76 0.00 459.63 -0.0702 -0.0517 -0.0418 0.016822 110.00 14.87 478.71 0.00 457.62 -0.0702 -0.0527 -0.0415 0.016723 110.00 14.51 426.67 0.00 452.62 -0.0702 -0.0551 -0.0407 0.016624 110.00 14.09 363.91 0.00 446.60 -0.0702 -0.0580 -0.0399 0.0164

51

Table 10: DC OPF Solution Results in SI Units for Five-Node Case - LMP Values (EqualityConstraint Multipliers) and Minimized Total Variable Cost

Hour LMP1 LMP2 LMP3 LMP4 LMP5 minTVC01 15.17 35.50 31.65 21.05 16.21 19587.1102 15.16 33.95 30.39 20.60 16.13 17107.2503 15.16 32.92 29.55 20.30 16.07 15556.7504 15.16 32.40 29.13 20.15 16.04 14800.9305 15.15 31.89 28.72 20.00 16.01 14076.0906 15.16 32.15 28.93 20.07 16.03 14436.4807 15.16 32.40 29.13 20.15 16.04 14800.9308 15.16 33.44 29.97 20.45 16.10 16330.2009 15.17 36.01 32.06 21.20 16.24 20433.8810 15.18 38.08 33.74 21.81 16.35 24043.6311 15.18 38.60 34.16 21.96 16.38 24993.9012 15.18 38.85 34.37 22.03 16.39 25467.4713 15.18 38.60 34.16 21.96 16.38 24993.9014 15.18 38.08 33.74 21.81 16.35 24043.6315 15.17 37.82 33.53 21.73 16.34 23583.1016 15.17 37.82 33.53 21.73 16.34 23583.1017 15.18 38.85 34.37 22.03 16.39 25467.4718 14.02 78.24 66.07 32.61 17.32 31038.5119 15.07 45.55 39.78 23.90 16.64 28006.8820 15.18 39.88 35.20 22.33 16.45 27422.3721 15.18 39.63 35.00 22.26 16.43 26931.8922 15.18 39.11 34.57 22.11 16.41 25945.8523 15.17 37.82 33.53 21.73 16.34 23583.1024 15.17 36.28 32.28 21.28 16.25 20879.49

52

Table 11: DC OPF Solution Results in SI Units for Five-Node Case - Thermal Limit In-equality Constraint Multipliers for Each Branch in Each Direction (km and mk)

Branch km multipliersHour 12 14 15 23 34 45 21 41 51 32 43 5401 30.36 0 0 0 0 0 0 0 0 0 0 002 28.05 0 0 0 0 0 0 0 0 0 0 003 26.52 0 0 0 0 0 0 0 0 0 0 004 25.74 0 0 0 0 0 0 0 0 0 0 005 24.99 0 0 0 0 0 0 0 0 0 0 006 25.37 0 0 0 0 0 0 0 0 0 0 007 25.74 0 0 0 0 0 0 0 0 0 0 008 27.29 0 0 0 0 0 0 0 0 0 0 009 31.12 0 0 0 0 0 0 0 0 0 0 010 34.20 0 0 0 0 0 0 0 0 0 0 011 34.97 0 0 0 0 0 0 0 0 0 0 012 35.35 0 0 0 0 0 0 0 0 0 0 013 34.97 0 0 0 0 0 0 0 0 0 0 014 34.20 0 0 0 0 0 0 0 0 0 0 015 33.82 0 0 0 0 0 0 0 0 0 0 016 33.82 0 0 0 0 0 0 0 0 0 0 017 35.35 0 0 0 0 0 0 0 0 0 0 018 95.88 0 0 0 0 0 0 0 0 0 0 019 45.50 0 0 0 0 0 0 0 0 0 0 020 36.88 0 0 0 0 0 0 0 0 0 0 021 36.50 0 0 0 0 0 0 0 0 0 0 022 35.73 0 0 0 0 0 0 0 0 0 0 023 33.82 0 0 0 0 0 0 0 0 0 0 024 31.51 0 0 0 0 0 0 0 0 0 0 0

53

Table 12: DC OPF Solution Results in SI Units for Five-Node Case - Lower and UpperProduction Inequality Constraint Multipliers for Each Generator

Hour P LG1 P L

G2 P LG3 P L

G4 P LG5 P U

G1 P UG2 P U

G3 P UG4 P U

G5

01 0 0 0 8.95 0 0.07 0 0 0 002 0 0 0 9.40 0 0.06 0 0 0 003 0 0 0 9.70 0 0.06 0 0 0 004 0 0 0 9.85 0 0.06 0 0 0 005 0 0 0 10.00 0 0.05 0 0 0 006 0 0 0 9.93 0 0.06 0 0 0 007 0 0 0 9.85 0 0.06 0 0 0 008 0 0 0 9.55 0 0.06 0 0 0 009 0 0 0 8.80 0 0.07 0 0 0 010 0 0 0 8.19 0 0.08 0 0 0 011 0 0 0 8.04 0 0.08 0 0 0 012 0 0 0 7.97 0 0.08 0 0 0 013 0 0 0 8.04 0 0.08 0 0 0 014 0 0 0 8.19 0 0.08 0 0 0 015 0 0 0 8.27 0 0.07 0 0 0 016 0 0 0 8.27 0 0.07 0 0 0 017 0 0 0 7.97 0 0.08 0 0 0 018 0 0.98 0 0 0 0 0 30.67 0 019 0 0 0 6.10 0 0 0 4.38 0 020 0 0 0 7.67 0 0.08 0 0 0 021 0 0 0 7.74 0 0.08 0 0 0 022 0 0 0 7.89 0 0.08 0 0 0 023 0 0 0 8.27 0 0.07 0 0 0 024 0 0 0 8.72 0 0.07 0 0 0 0

54

Table 13: DC OPF Solution Results in SI Units for Five-Node Case - Optimal Real PowerBranch Flow Pkm and Its Associated Thermal Limit P U

km for Each km ∈ BI

Hour P12a P14 P15 P23 P34 P45

01 250.00 129.65 -255.77 -100.00 -67.47 -187.8202 250.00 126.71 -253.27 -72.93 -80.32 -184.2703 250.00 124.77 -251.61 -55.04 -88.81 -181.9304 250.00 123.79 -250.77 -46.02 -93.09 -180.7405 250.00 122.83 -249.95 -37.16 -97.30 -179.5806 250.00 123.31 -250.36 -41.59 -95.19 -180.1607 250.00 123.79 -250.77 -46.02 -93.09 -180.7408 250.00 125.75 -252.45 -64.07 -84.52 -183.1109 250.00 130.61 -256.60 -108.86 -63.26 -188.9810 250.00 134.51 -259.92 -144.80 -46.20 -193.6911 250.00 135.49 -260.76 -153.82 -41.92 -194.8712 250.00 135.97 -261.17 -158.25 -39.81 -195.4513 250.00 135.49 -260.76 -153.82 -41.92 -194.8714 250.00 134.51 -259.92 -144.80 -46.20 -193.6915 250.00 134.03 -259.51 -140.37 -48.30 -193.1116 250.00 134.03 -259.51 -140.37 -48.30 -193.1117 250.00 135.97 -261.17 -158.25 -39.81 -195.4518 250.00 98.83 -346.76 -198.62 -63.15 -175.8819 250.00 137.64 -274.17 -180.73 -29.93 -199.9620 250.00 137.91 -262.83 -176.14 -31.32 -197.8021 250.00 137.43 -262.42 -171.71 -33.42 -197.2222 250.00 136.45 -261.58 -162.69 -37.71 -196.0323 250.00 134.03 -259.51 -140.37 -48.30 -193.1124 250.00 131.11 -257.02 -113.46 -61.08 -189.58

P U12 P U

14 P U15 P U

23 P U34 P U

45

250.00 150.00 400.00 350.00 240.00 240.00

aIn accordance with the usual convention, the real power Pkm flowing along a branch km is positivelyvalued if and only if real power is flowing from node k to node m.

55

Table 14: Sensitivity Test Results for Three-Node Case (π = 100, Angles in Radians)

Hour p∗G1 p∗G2 p∗G3 δ∗2 δ∗3 LMP1 LMP2 LMP3 minTVC01 200.0 16.1 5.0 -0.079920 -0.109520 18.2555 18.2971 18.3239 2993.9502 189.0 10.0 5.0 -0.080800 -0.104800 12.4441 12.4858 12.5094 2724.3303 177.7 10.0 5.0 -0.075216 -0.097887 12.3395 12.3783 12.4005 2565.1204 172.0 10.0 5.0 -0.072400 -0.094400 12.2867 12.3240 12.3455 2485.7005 166.4 10.0 5.0 -0.069633 -0.090974 12.2349 12.2708 12.2915 2408.2706 169.2 10.0 5.0 -0.071016 -0.092687 12.2608 12.2974 12.3185 2446.9107 172.0 10.0 5.0 -0.072400 -0.094400 12.2867 12.3240 12.3455 2485.7008 183.4 10.0 5.0 -0.078033 -0.101374 12.3923 12.4325 12.4554 2645.1309 200.0 21.7 5.0 -0.074122 -0.107675 18.3266 18.3656 18.3941 3097.9010 200.0 44.4 5.0 -0.050621 -0.100198 18.6149 18.6435 18.6786 3527.1311 200.0 50.1 5.0 -0.044720 -0.098320 18.6873 18.7132 18.7500 3636.9012 200.0 52.9 5.0 -0.041821 -0.097398 18.7229 18.7475 18.7851 3691.1113 200.0 50.1 5.0 -0.044720 -0.098320 18.6873 18.7132 18.7500 3636.9014 200.0 44.4 5.0 -0.050621 -0.100198 18.6149 18.6435 18.6786 3527.1315 200.0 41.6 5.0 -0.053520 -0.101120 18.5794 18.6092 18.6435 3473.5116 200.0 41.6 5.0 -0.053520 -0.101120 18.5794 18.6092 18.6435 3473.5117 200.0 52.9 5.0 -0.041821 -0.097398 18.7229 18.7475 18.7851 3691.1118 200.0 78.4 5.0 -0.015421 -0.088998 19.0468 19.0596 19.1047 4193.6419 200.0 67.1 5.0 -0.027120 -0.092720 18.9033 18.9213 18.9631 3968.9820 200.0 64.2 5.0 -0.030122 -0.093675 18.8664 18.8858 18.9267 3911.8321 200.0 61.4 5.0 -0.033021 -0.094598 18.8309 18.8515 18.8916 3856.8522 200.0 55.7 5.0 -0.038922 -0.096475 18.7585 18.7818 18.8202 3745.5123 200.0 41.6 5.0 -0.053520 -0.101120 18.5794 18.6092 18.6435 3473.5124 200.0 24.6 5.0 -0.071120 -0.106720 18.3634 18.4011 18.4304 3152.03

56

Table 15: Sensitivity Test Results for Three-Node Case (π = 10, Angles in Radians)

Hour p∗G1 p∗G2 p∗G3 δ∗2 δ∗3 LMP1 LMP2 LMP3 minTVC01 200.0 16.1 5.0 -0.079920 -0.109520 18.2929 18.2971 18.2997 2993.9502 189.0 10.0 5.0 -0.080800 -0.104800 12.4441 12.4483 12.4507 2724.3303 177.7 10.0 5.0 -0.075216 -0.097887 12.3395 12.3434 12.3456 2565.1204 172.0 10.0 5.0 -0.072400 -0.094400 12.2867 12.2905 12.2926 2485.7005 166.4 10.0 5.0 -0.069633 -0.090974 12.2349 12.2385 12.2405 2408.2706 169.2 10.0 5.0 -0.071016 -0.092687 12.2608 12.2645 12.2666 2446.9107 172.0 10.0 5.0 -0.072400 -0.094400 12.2867 12.2905 12.2926 2485.7008 183.4 10.0 5.0 -0.078033 -0.101374 12.3923 12.3963 12.3986 2645.1309 200.0 21.7 5.0 -0.074122 -0.107675 18.3617 18.3656 18.3685 3097.9010 200.0 44.4 5.0 -0.050621 -0.100198 18.6406 18.6435 18.6470 3527.1311 200.0 50.1 5.0 -0.044720 -0.098320 18.7106 18.7132 18.7169 3636.9012 200.0 52.9 5.0 -0.041821 -0.097398 18.7450 18.7475 18.7513 3691.1113 200.0 50.1 5.0 -0.044720 -0.098320 18.7106 18.7132 18.7169 3636.9014 200.0 44.4 5.0 -0.050621 -0.100198 18.6406 18.6435 18.6470 3527.1315 200.0 41.6 5.0 -0.053520 -0.101120 18.6062 18.6092 18.6126 3473.5116 200.0 41.6 5.0 -0.053520 -0.101120 18.6062 18.6092 18.6126 3473.5117 200.0 52.9 5.0 -0.041821 -0.097398 18.7450 18.7475 18.7513 3691.1118 200.0 78.4 5.0 -0.015421 -0.088998 19.0583 19.0596 19.0641 4193.6419 200.0 67.1 5.0 -0.027120 -0.092720 18.9195 18.9213 18.9255 3968.9820 200.0 64.2 5.0 -0.030122 -0.093675 18.8839 18.8858 18.8899 3911.8321 200.0 61.4 5.0 -0.033021 -0.094598 18.8495 18.8515 18.8555 3856.8522 200.0 55.7 5.0 -0.038922 -0.096475 18.7794 18.7818 18.7856 3745.5123 200.0 41.6 5.0 -0.053520 -0.101120 18.6062 18.6092 18.6126 3473.5124 200.0 24.6 5.0 -0.071120 -0.106720 18.3973 18.4011 18.4040 3152.03

57

Table 16: Sensitivity Test Results for Three-Node Case (π = 1, Angles in Radians)

Hour p∗G1 p∗G2 p∗G3 δ∗2 δ∗3 LMP1 LMP2 LMP3 minTVC01 200.0 16.1 5.0 -0.079920 -0.109520 18.2966 18.2971 18.2973 2993.9502 189.0 10.0 5.0 -0.080800 -0.104800 12.4441 12.4446 12.4448 2724.3303 177.7 10.0 5.0 -0.075216 -0.097887 12.3395 12.3399 12.3401 2565.1204 172.0 10.0 5.0 -0.072400 -0.094400 12.2867 12.2871 12.2873 2485.7005 166.4 10.0 5.0 -0.069633 -0.090974 12.2349 12.2352 12.2354 2408.2706 169.2 10.0 5.0 -0.071016 -0.092687 12.2608 12.2612 12.2614 2446.9107 172.0 10.0 5.0 -0.072400 -0.094400 12.2867 12.2871 12.2873 2485.7008 183.4 10.0 5.0 -0.078033 -0.101374 12.3923 12.3927 12.3929 2645.1309 200.0 21.7 5.0 -0.074122 -0.107675 18.3652 18.3656 18.3659 3097.9010 200.0 44.4 5.0 -0.050621 -0.100198 18.6432 18.6435 18.6438 3527.1311 200.0 50.1 5.0 -0.044720 -0.098320 18.7130 18.7132 18.7136 3636.9012 200.0 52.9 5.0 -0.041821 -0.097398 18.7473 18.7475 18.7479 3691.1113 200.0 50.1 5.0 -0.044720 -0.098320 18.7130 18.7132 18.7136 3636.9014 200.0 44.4 5.0 -0.050621 -0.100198 18.6432 18.6435 18.6438 3527.1315 200.0 41.6 5.0 -0.053520 -0.101120 18.6089 18.6092 18.6095 3473.5116 200.0 41.6 5.0 -0.053520 -0.101120 18.6089 18.6092 18.6095 3473.5117 200.0 52.9 5.0 -0.041821 -0.097398 18.7473 18.7475 18.7479 3691.1118 200.0 78.4 5.0 -0.015421 -0.088998 19.0595 19.0596 19.0601 4193.6419 200.0 67.1 5.0 -0.027120 -0.092720 18.9211 18.9213 18.9217 3968.9820 200.0 64.2 5.0 -0.030122 -0.093675 18.8856 18.8858 18.8862 3911.8321 200.0 61.4 5.0 -0.033021 -0.094598 18.8513 18.8515 18.8519 3856.8522 200.0 55.7 5.0 -0.038922 -0.096475 18.7815 18.7818 18.7822 3745.5123 200.0 41.6 5.0 -0.053520 -0.101120 18.6089 18.6092 18.6095 3473.5124 200.0 24.6 5.0 -0.071120 -0.106720 18.4007 18.4011 18.4014 3152.03

58

Table 17: Sensitivity Test Results for Three-Node Case (π = 0.1, Angles in Radians)

Hour p∗G1 p∗G2 p∗G3 δ∗2 δ∗3 LMP1 LMP2 LMP3 minTVC01 200.0 16.1 5.0 -0.079920 -0.109520 18.2970 18.2971 18.2971 2993.9502 189.0 10.0 5.0 -0.080800 -0.104800 12.4441 12.4442 12.4442 2724.3303 177.7 10.0 5.0 -0.075216 -0.097887 12.3395 12.3395 12.3396 2565.1204 172.0 10.0 5.0 -0.072400 -0.094400 12.2867 12.2868 12.2868 2485.7005 166.4 10.0 5.0 -0.069633 -0.090974 12.2349 12.2349 12.2349 2408.2706 169.2 10.0 5.0 -0.071016 -0.092687 12.2608 12.2608 12.2608 2446.9107 172.0 10.0 5.0 -0.072400 -0.094400 12.2867 12.2868 12.2868 2485.7008 183.4 10.0 5.0 -0.078033 -0.101374 12.3923 12.3923 12.3923 2645.1309 200.0 21.7 5.0 -0.074122 -0.107675 18.3656 18.3656 18.3656 3097.9010 200.0 44.4 5.0 -0.050621 -0.100198 18.6434 18.6435 18.6435 3527.1311 200.0 50.1 5.0 -0.044720 -0.098320 18.7132 18.7132 18.7133 3636.9012 200.0 52.9 5.0 -0.041821 -0.097398 18.7475 18.7475 18.7475 3691.1113 200.0 50.1 5.0 -0.044720 -0.098320 18.7132 18.7132 18.7133 3636.9014 200.0 44.4 5.0 -0.050621 -0.100198 18.6434 18.6435 18.6435 3527.1315 200.0 41.6 5.0 -0.053520 -0.101120 18.6092 18.6092 18.6092 3473.5116 200.0 41.6 5.0 -0.053520 -0.101120 18.6092 18.6092 18.6092 3473.5117 200.0 52.9 5.0 -0.041821 -0.097398 18.7475 18.7475 18.7475 3691.1118 200.0 78.4 5.0 -0.015421 -0.088998 19.0596 19.0596 19.0597 4193.6419 200.0 67.1 5.0 -0.027120 -0.092720 18.9213 18.9213 18.9213 3968.9820 200.0 64.2 5.0 -0.030122 -0.093675 18.8858 18.8858 18.8858 3911.8321 200.0 61.4 5.0 -0.033021 -0.094598 18.8515 18.8515 18.8516 3856.8522 200.0 55.7 5.0 -0.038922 -0.096475 18.7817 18.7818 18.7818 3745.5123 200.0 41.6 5.0 -0.053520 -0.101120 18.6092 18.6092 18.6092 3473.5124 200.0 24.6 5.0 -0.071120 -0.106720 18.4011 18.4011 18.4011 3152.03

59

Table 18: Sensitivity Test Results for Three-Node Case (π = 0.01, Angles in Radians)

Hour p∗G1 p∗G2 p∗G3 δ∗2 δ∗3 LMP1 LMP2 LMP3 minTVC01 200.0 16.1 5.0 -0.079920 -0.109520 18.2971 18.2971 18.2971 2993.9502 189.0 10.0 5.0 -0.080800 -0.104800 12.4441 12.4441 12.4441 2724.3303 177.7 10.0 5.0 -0.075216 -0.097887 12.3395 12.3395 12.3395 2565.1204 172.0 10.0 5.0 -0.072400 -0.094400 12.2867 12.2867 12.2867 2485.7005 166.4 10.0 5.0 -0.069633 -0.090974 12.2349 12.2349 12.2349 2408.2706 169.2 10.0 5.0 -0.071016 -0.092687 12.2608 12.2608 12.2608 2446.9107 172.0 10.0 5.0 -0.072400 -0.094400 12.2867 12.2867 12.2867 2485.7008 183.4 10.0 5.0 -0.078033 -0.101374 12.3923 12.3923 12.3923 2645.1309 200.0 21.7 5.0 -0.074122 -0.107675 18.3656 18.3656 18.3656 3097.9010 200.0 44.4 5.0 -0.050621 -0.100198 18.6435 18.6435 18.6435 3527.1311 200.0 50.1 5.0 -0.044720 -0.098320 18.7132 18.7132 18.7132 3636.9012 200.0 52.9 5.0 -0.041821 -0.097398 18.7475 18.7475 18.7475 3691.1113 200.0 50.1 5.0 -0.044720 -0.098320 18.7132 18.7132 18.7132 3636.9014 200.0 44.4 5.0 -0.050621 -0.100198 18.6435 18.6435 18.6435 3527.1315 200.0 41.6 5.0 -0.053520 -0.101120 18.6092 18.6092 18.6092 3473.5116 200.0 41.6 5.0 -0.053520 -0.101120 18.6092 18.6092 18.6092 3473.5117 200.0 52.9 5.0 -0.041821 -0.097398 18.7475 18.7475 18.7475 3691.1118 200.0 78.4 5.0 -0.015421 -0.088998 19.0596 19.0596 19.0596 4193.6419 200.0 67.1 5.0 -0.027120 -0.092720 18.9213 18.9213 18.9213 3968.9820 200.0 64.2 5.0 -0.030122 -0.093675 18.8858 18.8858 18.8858 3911.8321 200.0 61.4 5.0 -0.033021 -0.094598 18.8515 18.8515 18.8515 3856.8522 200.0 55.7 5.0 -0.038922 -0.096475 18.7818 18.7818 18.7818 3745.5123 200.0 41.6 5.0 -0.053520 -0.101120 18.6092 18.6092 18.6092 3473.5124 200.0 24.6 5.0 -0.071120 -0.106720 18.4011 18.4011 18.4011 3152.03

60

Table 19: Sensitivity Test Results for Three-Node Case - Cross Comparison for Sum ofSquared Voltage Angle Differences for π = 100, 10, 1, 0.1, 0.01, Angles in Radians

Hour SSVAD100a SSVAD10 SSVAD1 SSVAD0.1 SSVAD0.01 MaxADb

01 0.019257997 0.019257997 0.019257997 0.019257997 0.019257997 8.10E-1502 0.018087680 0.018087680 0.018087680 0.018087680 0.018087680 4.16E-1403 0.015753349 0.015753349 0.015753349 0.015753349 0.015753349 3.95E-1404 0.014637120 0.014637120 0.014637120 0.014637120 0.014637120 3.83E-1405 0.013580482 0.013580482 0.013580482 0.013580482 0.013580482 3.71E-1406 0.014103844 0.014103844 0.014103844 0.014103844 0.014103844 3.77E-1407 0.014637120 0.014637120 0.014637120 0.014637120 0.014637120 3.83E-1408 0.016910662 0.016910662 0.016910662 0.016910662 0.016910662 4.05E-1409 0.018213892 0.018213892 0.018213892 0.018213892 0.018213892 7.30E-1510 0.015059898 0.015059898 0.015059898 0.015059898 0.015059898 4.50E-1511 0.014539661 0.014539661 0.014539661 0.014539661 0.014539661 3.80E-1512 0.014324057 0.014324057 0.014324057 0.014324057 0.014324057 3.80E-1513 0.014539661 0.014539661 0.014539661 0.014539661 0.014539661 3.80E-1514 0.015059898 0.015059898 0.015059898 0.015059898 0.015059898 4.50E-1515 0.015355405 0.015355405 0.015355405 0.015355405 0.015355405 4.70E-1516 0.015355405 0.015355405 0.015355405 0.015355405 0.015355405 4.70E-1517 0.014324057 0.014324057 0.014324057 0.014324057 0.014324057 3.80E-1518 0.013571891 0.013571891 0.013571891 0.013571891 0.013571891 1.19E-1419 0.013635853 0.013635853 0.013635853 0.013635853 0.013635853 1.50E-1420 0.013721393 0.013721393 0.013721393 0.013721393 0.013721393 1.63E-1421 0.013830775 0.013830775 0.013830775 0.013830775 0.013830775 1.76E-1422 0.014134773 0.014134773 0.014134773 0.014134773 0.014134773 1.99E-1423 0.015355405 0.015355405 0.015355405 0.015355405 0.015355405 4.70E-1524 0.017714573 0.017714573 0.017714573 0.017714573 0.017714573 7.00E-15

aSum of squared voltage angle differences for a specific choice of π, where π is specified to be 100, 10, 1,0.1 or 0.01. More precisely, SSVAD =

∑km∈BI[δ

∗k − δ∗m]2

bMaximum absolute difference between any two SSVAD values

61

Table 20: Sensitivity Test Results for Five-Node Case - Cross Comparison for Sum of SquaredVoltage Angle Differences for π = 100, 10, 1, 0.1, 0.01, Angles in Radians

Hour SSVAD100 SSVAD10 SSVAD1 SSVAD0.1 SSVAD0.01 MaxAD01 0.010386061 0.010386162 0.010386172 0.010386173 0.010386173 1.12E-0702 0.010307655 0.010307759 0.010307769 0.010307770 0.010307771 1.16E-0703 0.010283485 0.010283591 0.010283602 0.010283603 0.010283603 1.18E-0704 0.010279443 0.010279550 0.010279561 0.010279562 0.010279562 1.19E-0705 0.010280962 0.010281070 0.010281081 0.010281082 0.010281082 1.20E-0706 0.010279593 0.010279701 0.010279712 0.010279713 0.010279713 1.20E-0707 0.010279443 0.010279550 0.010279561 0.010279562 0.010279562 1.19E-0708 0.010292874 0.010292979 0.010292989 0.010292990 0.010292991 1.17E-0709 0.010422608 0.010422708 0.010422718 0.010422719 0.010422719 1.11E-0710 0.010625778 0.010625874 0.010625884 0.010625885 0.010625885 1.07E-0711 0.010690587 0.010690682 0.010690691 0.010690692 0.010690692 1.05E-0712 0.010724577 0.010724671 0.010724680 0.010724681 0.010724681 1.05E-0713 0.010690587 0.010690682 0.010690691 0.010690692 0.010690692 1.05E-0714 0.010625778 0.010625874 0.010625884 0.010625885 0.010625885 1.07E-0715 0.010595866 0.010595962 0.010595972 0.010595973 0.010595973 1.07E-0716 0.010595866 0.010595962 0.010595972 0.010595973 0.010595973 1.07E-0717 0.010724577 0.010724671 0.010724680 0.010724681 0.010724681 1.05E-0718 0.009870395 0.009870652 0.009870678 0.009870680 0.009870681 2.86E-0719 0.010980723 0.010980723 0.010980723 0.010980723 0.010980722 6.60E-1020 0.010875114 0.010875206 0.010875215 0.010875216 0.010875216 1.03E-0721 0.010835704 0.010835797 0.010835806 0.010835807 0.010835807 1.03E-0722 0.010759863 0.010759957 0.010759967 0.010759968 0.010759968 1.04E-0723 0.010595866 0.010595962 0.010595972 0.010595973 0.010595973 1.07E-0724 0.010443596 0.010443696 0.010443705 0.010443707 0.010443707 1.10E-07

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